The Basic Formulas Derivation and Degradation Verification of the 3-D Dynamic Elastoplastic TD-BEM

Abstract

In the field of dynamics research, in-depth exploration of three-dimensional (3-D) elastoplastic dynamics is crucial for understanding material behavior under complex dynamic loads. The findings hold significant guiding implications for design optimization in practical engineering domains such as aerospace and mechanical engineering. Current methodologies for solving 3-D dynamic elastoplastic problems face challenges: While traditional finite element methods (FEMs) excel in handling material nonlinearity, they encounter limitations in 3-D dynamic analysis, especially difficulties in simulating infinite domains. Although classical time-domain boundary element methods (TD-BEMs) effectively reduce computational dimensionality through dimension reduction and time-domain fundamental solutions, they remain underdeveloped for 3-D elastoplastic analysis. This study mainly includes the following contributions: First, we derived the 3-D dynamic elastoplastic boundary integral equations using the initial strain method for the first time, which aligns with the physical essence of strain decomposition in elastoplastic theory. Second, kernel functions for displacement, traction, and strain influence coefficients are analytically obtained by integrating time-domain fundamental solutions with physical and geometric equations. To validate the formulation, a 3-D-to-2-D transformation is implemented through an integral degradation method, converting the problem into a verified dynamic plane strain elastoplastic system.

1. Introduction

The finite element method (FEM) has long been the cornerstone of computational mechanics due to its adaptability in addressing material nonlinearity. However, its limitations become evident when tackling large-scale engineering challenges, particularly those involving infinite domains. FEM’s requirement for full-domain discretization necessitates artificial boundary truncations, introducing errors and computational inefficiencies. In contrast, the boundary element method (BEM) inherently resolves these issues through dimensionality reduction, discretizing only boundary surfaces and localized plastic zones. This approach not only minimizes mesh distortion sensitivity but also seamlessly incorporates infinite domains via its fundamental solution framework, eliminating the need for auxiliary virtual boundaries [1,2,3]. Such characteristics position BEM as a superior candidate for dynamic elastoplastic analysis, especially when combined with time-domain formulations.

The boundary integral equation of BEM can be established by the initial stress method and the initial strain method. Initial stress and initial strain are two aspects of the same physical nature. When dealing with practical problems, it is necessary to choose according to the convenience of calculation. For example, when dealing with the problem of temperature deformation, since temperature change directly causes the strain change of the material, the initial strain consideration can be more intuitive and simple to build the model and calculation because the strain increment caused by temperature change can be directly introduced into the analysis as the initial strain. When dealing with problems such as ground stress, geological structure and other factors make the initial stress state relatively clear, and it is more appropriate to consider the initial stress.

The use of the initial strain method for studying elastoplastic problems began with the publication of a paper by J.L. Swedlow and T.A. Cruse (1971), in which they proposed the first displacement boundary integral formula based on 3-D elastoplastic problems [4]. Although this formula was not used for specific problem analysis and did not provide integral expressions for stress or strain, it still had significant implications for the further development of the boundary element method in the field of elastoplasticity. Bui (1978) corrected errors in the previous literature and derived the correct expression for point strains inside 3-D objects, making important contributions to the development of elastoplastic boundary elements [5]. J.C.F. Telles and C.A. Brebbbia (1979) presented boundary element formulas for 2-D and 3-D elastoplastic problems, as well as providing correct expressions for internal stresses [6]. The initial stress method was proposed by Chaudonneret (1977) as well as Banerjee PK and Cathie (1978), further refining the theory of the boundary element method [7,8].

The development of time-domain boundary element methods (TD-BEMs) has been pivotal in advancing dynamics [9,10,11,12]. While Ahmad (1990) pioneered TD-BEM for 3-D dynamic elastoplasticity [13], Li et al. (2021) notably established a rigorous 2-D plane strain framework using the initial strain method [14,15], demonstrating its superiority in capturing strain decomposition—a cornerstone of elastoplastic theory. Concurrently, dimensional reduction techniques emerged as a validation paradigm. Studies by Niwa and Manolis [16,17] demonstrated that 3-D-fundamental solutions could be systematically degraded to 2-D plane strain systems, such as modeling 2-D domains as elongated cylindrical volumes. These efforts confirmed the consistency between 2-D and 3-D formulations, setting a precedent for verifying higher-dimensional theories through their lower-dimensional counterparts.

Despite these advancements, a critical gap persists: the initial strain method remains underexplored for 3-D dynamic elastoplastic analysis. Existing TD-BEM frameworks predominantly rely on initial stress formulations, which, while computationally tractable, lack direct physical correspondence to the strain-driven nature of plastic deformation. This study bridges this gap through three contributions:

• The basic 3-D formulas derivation: For the first time, a complete set of 3-D dynamic elastoplastic boundary integral equations is derived using the initial strain method. The initial strain method directly reflects physical essence from the perspective of the strain and also provides a more convenient way for the subsequent mathematical treatment and equation construction.
• Analytical Kernel Development: Displacement, traction, and strain influence kernel functions are analytically developed by integrating time-domain fundamental solutions with physical and geometric equations.
• Integral Degradation: Building on historical precedents [16,17,18,19,20,21,22,23,24,25], this work employs a systematic 3-D-to-2-D integral degradation strategy to validate the proposed formulation. By constraining out-of-plane displacements and isolating third-direction strain contributions, the 3-D equations are reduced to a 2-D plane strain system. The degenerated solutions exhibit equivalence to established 2-D formulations [14], confirming the correctness and universality of the 3-D framework.

The rationale for dimensional reduction lies in its proven efficacy as a validation tool. Earlier studies demonstrated that 3-D fundamental solutions, when integrated along constrained axes, reproduce 2-D results [13,14,15]. Therefore, the mature 2-D system can also be used as a reference to verify the correctness of the 3-D derivation.

In summary, within the development history of the TD-BEM, the initial stress method has effectively addressed 2-D and 3-D elastoplastic dynamic problems, achieving important breakthroughs through solving corresponding stresses via integral equations. However, there remains a gap in utilizing the initial strain method for solving 3-D dynamic elastoplastic problems, requiring further improvement in related theoretical research. For the first time, the initial strain method is used to establish the TD-BEM formula for 3-D dynamic elastoplastic analysis in this study. Subsequently, the time-domain boundary integral equation and influence coefficient kernel function of 3-D dynamic elastoplastic problems are derived. And the 3-D fundamental solution is degraded to verify the correctness of the derivation.

2. Time-Domain Boundary Integral Equations for 3-D Elastoplastic Dynamic Problems

Under the conditions of zero initial condition and negligible body force, the displacement boundary integral equation of 3-D elastoplastic dynamics can be expressed in the space-time domain by using the Graffi’s dynamic reciprocity theorem [26], as shown in Equation (1).

$$\begin{array}{l}{c}_{ik}{u}_i\left(P,t\right)=-{\displaystyle {\int }_S{\displaystyle {\int }_0^t{p}_{ik}^{*}\left(P,\tau ;Q,t\right)}}{u}_k\left(Q,\tau \right)d\tau dS\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.33em}+{\displaystyle {\int }_S{\displaystyle {\int }_0^t{u}_{ik}^{*}\left(P,\tau ;Q,t\right)}}{p}_k\left(Q,\tau \right)d\tau dS\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.33em}+{\displaystyle {\int }_V{\displaystyle {\int }_0^t{\sigma }_{ikl}^{*}\left(P,\tau ;R,t\right)}}{\epsilon }_{kl}^p\left(R,\tau \right)d\tau dV\end{array}$$

(1)

where, i, k, l = 1, 2, 3; $c_{ik}$ is the position coefficient of the source point P [27], which is expressed as Equation (2); $p_{ik}^{*}$ and $u_{ik}^{*}$ are the 3-D time-domain fundamental solutions for the surface traction and displacement, respectively, as shown in Equations (4) and (5), which respectively represent the surface traction and displacement component of field point Q at t time x_k direction, caused by the unit concentrated force pulse applied to the source point P at τ time x_i direction; ${\sigma }_{ikl}^{*}$ is fundamental solution for stress, as shown in Equation (7), which represents the stress component of field point R at t time x_k direction in the plane, with n_l as the normal vector caused by the unit impulse applied to source point P at τ time x_i direction.

$$c_{ik}=\left\{\begin{array}{l}{\delta }_{ik}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}P\in V\\ {\displaystyle \frac{1}{2}}{\delta }_{ik}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}P\in S\left(Smooth boundary\right)\\ {\delta }_{ik}+\underset{\epsilon \to 0}{\lim }{\displaystyle {\int }_{S_{\epsilon }}{p}_{ik}dS}\hspace{1em}P\in S\left(Non-smooth boundary\right)\end{array}\right.$$

(2)

where p_ik $p_{ik}$ is the surface traction fundamental solution for 3-D elastoplastic problem as follows [28]:

$$p_{ik}(P;Q)=-{\displaystyle \frac{1}{8\pi (1-\upsilon )r^2}}\left\{\left[(1-2\upsilon ){\delta }_{ik}+3r_{,i}{r}_{,k}\right]{\displaystyle \frac{\partial r}{\partial n}}-(1-2\upsilon )(r_{,i}{n}_k-r_{,k}{n}_i)\right\}$$

(3)

$p_{ik}^{*}$ and $u_{ik}^{*}$ are expressed by Equations (4) and (5) [29], respectively:

$$u_{ik}^{*}(P,\tau ;Q,t)={\displaystyle \frac{1}{4\pi \mu }}(\psi {\delta }_{ik}-\chi r_{,i}{r}_{,k})$$

(4)

$$p_{ik}^{*}(P,\tau ;Q,t)={\displaystyle \frac{1}{4\pi }}\left[\begin{array}{l}\left({\psi }_{,r}-{\displaystyle \frac{\chi }{r}}\right)\left({\displaystyle \frac{\partial r}{\partial n}}{\delta }_{ik}+r_{,k}{n}_i\right)-2 {\displaystyle \frac{\chi }{r}}\left(r_{,i}{n}_k-2r_{,i}{r}_{,k}{\displaystyle \frac{\partial r}{\partial n}}\right)\\ -2{\chi }_{,r^{r_{,i}}}{r}_{,k}{\displaystyle \frac{\partial r}{\partial n}}+\left({\displaystyle \frac{c_1^2}{c_2^2}}-2\right)\left({\psi }_{,r}-{\chi }_{,r}-2 {\displaystyle \frac{\chi }{r}}\right)r_{,i}{n}_k\end{array}\right]$$

(5)

${\sigma }_{ikl}^{*}$ has the following relationship with $p_{ik}^{*}$:

$$p_{ik}^{*}\left(P,\tau ;Q,t\right)={\sigma }_{ikl}^{*}\left(P,\tau ;Q,t\right)n_l$$

(6)

Therefore, ${\sigma }_{ikl}^{*}$ can be further expressed in Equation (7), as:

$${\sigma }_{ikl}^{*}={\displaystyle \frac{1}{4\pi }}\left[\begin{array}{l}\left({\psi }_{,r}-{\displaystyle \frac{\chi }{r}}\right)\left({\delta }_{ik}{r}_{,l}+{\delta }_{il}{r}_{,k}\right)-2{\displaystyle \frac{\chi }{r}}\left({\delta }_{kl}{r}_{,i}-2r_{,i}{r}_{,k}{r}_{,l}\right)\\ -2{\chi }_{,r}{r}_{,i}{r}_{,k}{r}_{,l}+\left({\displaystyle \frac{c_1^2}{c_2^2}}-2\right)\left({\psi }_{,r}-{\chi }_{,r}-2{\displaystyle \frac{\chi }{r}}\right){\delta }_{kl}{r}_{,i}\end{array}\right]$$

(7)

where the parameters in the above time-domain fundamental solutions are expressed as:

$$\psi ={\displaystyle \frac{c_2^2}{r^3}}{t}^{\prime } \left(H_2-H_1\right)+{\displaystyle \frac{1}{r}}{\delta }_2$$

(8)

$$\chi =3\psi -{\displaystyle \frac{2}{r}}{\delta }_2-{\displaystyle \frac{c_2^2}{c_1^2}}{\displaystyle \frac{1}{r}}{\delta }_1$$

(9)

$${\psi }_{,r}=-{\displaystyle \frac{\chi }{r}}-{\displaystyle \frac{1}{r^2}}\left({\delta }_2+{\displaystyle \frac{r}{c_2}}{\dot{\delta }}_1\right)$$

(10)

$${\chi }_{,r}=-{\displaystyle \frac{3\chi }{r}}-{\displaystyle \frac{1}{r^2}}\left({\delta }_2+{\displaystyle \frac{r}{c_2}}{\dot{\delta }}_2\right)+{\displaystyle \frac{c_2^2}{c_1^2}}{\displaystyle \frac{1}{r^2}}\left({\delta }_1+{\displaystyle \frac{r}{c_1}}{\dot{\delta }}_1\right)$$

(11)

$${\psi }_{,rr}={\displaystyle \frac{4\chi }{r^2}}+{\displaystyle \frac{1}{r^3}}\left(3{\delta }_2+{\displaystyle \frac{3r}{c_2}}{\dot{\delta }}_2+{\displaystyle \frac{r^2}{c_2^2}}{\ddot{\delta }}_2\right)+{\displaystyle \frac{c_2^2}{c_1^2}}{\displaystyle \frac{1}{r^3}}\left({\delta }_1+{\displaystyle \frac{r}{c_2}}{\dot{\delta }}_1\right)$$

(12)

$${\chi }_{,rr}={\displaystyle \frac{12\chi }{r^2}}+{\displaystyle \frac{1}{r^3}}\left(5{\delta }_2+{\displaystyle \frac{5r}{c_2}}{\dot{\delta }}_2+{\displaystyle \frac{r^2}{c_2^2}}{\ddot{\delta }}_2\right)-{\displaystyle \frac{c_2^2}{c_1^2}}{\displaystyle \frac{1}{r^3}}\left(5{\delta }_1+{\displaystyle \frac{5r}{c_2}}{\dot{\delta }}_1+{\displaystyle \frac{r^2}{c_2^2}}{\ddot{\delta }}_1\right)$$

(13)

$$\left\{\begin{array}{l}{H}_w=H\left(t^{\prime }-r/c_w\right)w=1,2\\ {\delta }_w=\delta \left(t^{\prime }-r/c_w\right)w=1,2\\ r_{,i}={\displaystyle \frac{\partial r}{\partial x_i^Q}}=-{\displaystyle \frac{\partial r}{\partial x_i^P}}={\displaystyle \frac{r_i}{r}}\\ r_i=x_i^Q-x_i^P\\ {\displaystyle \frac{\partial r}{\partial n}}=r_{,i}{n}_i\\ n_i=-{\displaystyle \frac{x_i}{n}}\end{array}\right.$$

(14)

In Equations (8)–(14), $t^{\prime }=t-\tau$, r is the distance from the source point P to the field point Q, $H\left(t^{\prime }-r/c_w\right)$, $\delta \left(t^{\prime }-r/c_w\right)$, and $\stackrel{·}{\delta }\left(t^{\prime }-r/c_w\right)$ (w = 1, 2) represent the Heaviside function, the Delta function, and the Delta function derivative, respectively. $c_1=\sqrt{(\lambda +2\mu )/\rho }$ and $c_2=\sqrt{\mu /\rho }$ are P-wave and S-wave velocities, μ and λ are Lamay’s constants, and ρ is the material density. So far, all the 3-D time-domain fundamental solutions required by the 3-D elastoplastic dynamic TD-BEM using the initial strain method have been deduced, and the kernel function of the time-domain influence coefficient is further derived by establishing the boundary integral equation of the internal point stress.

The derivation of these fundamental solutions is a critical step in ensuring the accuracy and stability of the boundary integral equations. By analytically expressing the displacement, traction, and stress solutions, this method avoids the numerical errors that can arise from discretization and approximation. This is particularly important in dynamic analyses, where small errors can accumulate over time and lead to significant deviations from the true solution.

3. Establishment of the Stress Boundary Integral Equation at Interior Points and Derivation of Kernel Function of the Time-Domain Influence Coefficient

For 3-D elastoplastic dynamic analysis, the unknowns cannot be solved independently by the displacement boundary integral equations. In the process of solving the unknowns, it is necessary to combine the physical equation and the geometric equation to obtain the stress boundary integral equation at the interior point.

The physical equation is:

$${\sigma }_{ij}\left(P,t\right)={\sigma }_{ij}^e\left(P,t\right)-{\sigma }_{ij}^p\left(P,t\right)$$

(15)

where the imaginary elastic stress and plastic stress, ${\sigma }_{ij}^e$ and ${\sigma }_{ij}^p$, are expressed as:

$$\left\{\begin{array}{l}{\sigma }_{ij}^e\left(P,t\right)=\lambda {\delta }_{ij}{\epsilon }_{mm}\left(P,t\right)+2\mu {\epsilon }_{ij}\left(P,t\right)\\ {\sigma }_{ij}^p\left(P,t\right)=\lambda {\delta }_{ij}{\epsilon }_{mm}^p\left(P,t\right)+2\mu {\epsilon }_{ij}^p\left(P,t\right)\end{array}\right.$$

(16)

The differential relationship between the total stress and the total strain is expressed in the geometric equation as follows:

$${\epsilon }_{ij}\left(P,t\right)={\displaystyle \frac{1}{2}}\left[u_{i,j}\left(P,t\right)+u_{j,i}\left(P,t\right)\right]$$

(17)

$u_{i,j}\left(X\left(P\right),t\right)$ and $u_{j,i}\left(X\left(P\right),t\right)$ represent the derivative with respect to the coordinates of the source point in a geometric equation. This means that the new derivatives that appear in the above equation are related to the coordinates of the source point. By substituting the displacement boundary integral equation at the interior point into Equation (17) and combining the physical equation with the geometric equation, the stress boundary integral equation at the interior point can be obtained and expressed as:

$$\begin{array}{l}{\sigma }_{ij}\left(P,t\right)=-{\displaystyle {\int }_S{\displaystyle {\int }_0^t{s}_{ijk}^{*}\left(P,\tau ;Q,t\right)u_k\left(Q,\tau \right)\mathrm{d}\tau \mathrm{d}S}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.33em}+{\displaystyle {\int }_S{\displaystyle {\int }_0^t{d}_{ijk}^{*}\left(P,\tau ;Q,t\right)p_k\left(Q,\tau \right)\mathrm{d}\tau \mathrm{d}S}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.33em}+{\displaystyle {\int }_V{\displaystyle {\int }_0^t{\sigma }_{ijkl}^{*}\left(P,\tau ;R,t\right){\epsilon }_{kl}^{\mathrm{p}}\left(R,\tau \right)\mathrm{d}\tau \mathrm{d}V}}-{\sigma }_{ij}^{\mathrm{p}}\end{array}$$

(18)

In Equation (18), ${\sigma }_{ij}$ is the stress of source point P, and $s_{ijk}^{*}$, $d_{ijk}^{*}$, and ${\sigma }_{ijkl}^{*}$ are the kernel functions of the influence coefficients derived from the time-domain fundamental solutions of displacement, traction, and strain, respectively, calculated by the following Equation (19):

$$\left\{\begin{array}{l}{s}_{ijk}^{*}=\lambda {\delta }_{ij}{p}_{mk,m}^{*}+\mu \left(p_{ik,j}^{*}+p_{jk,i}^{*}\right)\\ d_{ijk}^{*}=\lambda {\delta }_{ij}{u}_{mk,m}^{*}+\mu \left(u_{ik,j}^{*}+u_{jk,i}^{*}\right)\\ {\sigma }_{ijkl}^{*}=\lambda {\delta }_{ij}{\sigma }_{mkl,m}^{*}+\mu \left({\sigma }_{ikl,j}^{*}+{\sigma }_{jkl,i}^{*}\right)\end{array}\right.$$

(19)

where the specific expressions of $s_{ijk}^{*}$, $d_{ijk}^{*}$, and ${\sigma }_{ijkl}^{*}$ are:

$$\begin{array}{l}{s}_{ijk}^{*}={\displaystyle \frac{\mu }{4\pi }}\left\{{\displaystyle \frac{\partial r}{\partial n}}\left[4\left({\chi }_{,rr}-5{\displaystyle \frac{{\chi }_{,r}}{r}}+8{\displaystyle \frac{\chi }{r^2}}\right)r_{,i}{r}_{,j}{r}_{,k}\left({\psi }_{,rr}-{\displaystyle \frac{{\psi }_{,r}}{r}}-3{\displaystyle \frac{{\chi }_{,r}}{r}}+6{\displaystyle \frac{\chi }{r^2}}\right)\left({\delta }_{ik}{r}_{,j}+{\delta }_{jk}{r}_{,i}\right)\right.\right.\\ \hspace{1em}\hspace{1em}\left.+2\left(2{\displaystyle \frac{{\chi }_{,r}}{r}}-4{\displaystyle \frac{\chi }{r^2}}+{\displaystyle \frac{\lambda }{\mu }}\left({\chi }_{,rr}+{\displaystyle \frac{{\chi }_{,r}}{r}}-4{\displaystyle \frac{\chi }{r^2}}-{\psi }_{,rr}+{\displaystyle \frac{{\psi }_{,r}}{r}}\right)\right){\delta }_{ij}{r}_{,k}\right]\\ \hspace{1em}\hspace{1em}+2\left(2{\displaystyle \frac{{\chi }_{,r}}{r}}-4{\displaystyle \frac{\chi }{r^2}}+{\displaystyle \frac{\lambda }{\mu }}\left({\chi }_{,rr}+{\displaystyle \frac{{\chi }_{,r}}{r}}-4{\displaystyle \frac{\chi }{r^2}}-{\psi }_{,rr}+{\displaystyle \frac{{\psi }_{,r}}{r}}\right)\right)r_{,i}{r}_{,j}{n}_k\\ \hspace{1em}\hspace{1em}-\left({\psi }_{,rr}-{\displaystyle \frac{{\psi }_{,r}}{r}}-3{\displaystyle \frac{{\chi }_{,r}}{r}}+6{\displaystyle \frac{\chi }{r^2}}\right)r_{,k}\left(n_j{r}_{,i}+r_{,j}{n}_i\right)-2\left({\displaystyle \frac{{\psi }_{,r}}{r}}-{\displaystyle \frac{\chi }{r^2}}\right)\left({\delta }_{jk}{n}_i+{\delta }_{ik}{n}_j\right)\\ \left.\hspace{1em}\hspace{1em}+\left(4{\displaystyle \frac{\chi }{r^2}}+4{\displaystyle \frac{\lambda }{\mu }}\left({\displaystyle \frac{{\chi }_{,r}}{r}}+2{\displaystyle \frac{\chi }{r^2}}-{\displaystyle \frac{{\psi }_{,r}}{r}}\right)+{\displaystyle \frac{{\lambda }^2}{{\mu }^2}}\left({\chi }_{,rr}+4{\displaystyle \frac{{\chi }_{,r}}{r}}+2{\displaystyle \frac{\chi }{r^2}}-{\psi }_{,rr}-{\displaystyle \frac{{\psi }_{,r}}{r}}\right)\right)\left({\delta }_{ij}{n}_k\right)\right\}\end{array}$$

(20)

$$d_{ijk}^{*}={\displaystyle \frac{1}{4\pi }}\left[\begin{array}{l}\left({\displaystyle \frac{\chi }{r}}-{\psi }_{,r}\right)\left({\delta }_{ik}{r}_{,j}+{\delta }_{kj}{r}_{,i}\right)-2{\displaystyle \frac{\chi }{r}}\left(2r_{,i}{r}_{,j}{r}_{,k}-{\delta }_{ij}{r}_{,k}\right)\\ +2{\chi }_{,r}{r}_{,i}{r}_{,j}{r}_{,k}+\left({\displaystyle \frac{c_1^2}{c_2^2}}-2\right)\left({\chi }_{,r}+2{\displaystyle \frac{\chi }{r}}-{\psi }_{,r}\right){\delta }_{ij}{r}_{,k}\end{array}\right]$$

(21)

$${\sigma }_{ijkl}^{*}={\displaystyle \frac{\mu }{4\pi }}\left[\begin{array}{l}{S}_2\left(4r_{,i}{r}_{,j}{r}_{,k}{r}_{,l}-{\delta }_{il}{r}_{,j}{r}_{,k}-{\delta }_{ik}{r}_{,j}{r}_{,l}-{\delta }_{jl}{r}_{,i}{r}_{,k}-{\delta }_{jk}{r}_{,i}{r}_{,l}\right)\\ +{\displaystyle \frac{2}{r}}\left({\chi }_{,r}-{\displaystyle \frac{\chi }{r}}\right)\left(\begin{array}{l}2{\delta }_{jl}{r}_{,i}{r}_{,k}+2{\delta }_{jk}{r}_{,i}{r}_{,l}+2{\delta }_{kl}{r}_{,i}{r}_{,j}+2{\delta }_{il}{r}_{,j}{r}_{,k}\\ +2{\delta }_{ik}{r}_{,j}{r}_{,l}+2{\delta }_{ij}{r}_{,l}{r}_{,k}-16r_{,i}{r}_{,j}{r}_{,k}{r}_{,l}\end{array}\right)\\ +{\displaystyle \frac{1}{r}}\left({\psi }_{,r}-{\displaystyle \frac{\chi }{r}}\right)\left(\begin{array}{l}{\delta }_{jl}{r}_{,i}{r}_{,k}+{\delta }_{il}{r}_{,j}{r}_{,k}+{\delta }_{jk}{r}_{,i}{r}_{,l}+{\delta }_{ik}{r}_{,j}{r}_{,l}-\\ {\delta }_{il}{\delta }_{jk}-{\delta }_{jl}{\delta }_{ik}-{\delta }_{jk}{\delta }_{il}-{\delta }_{ik}{\delta }_{jl}\end{array}\right)\\ -{\displaystyle \frac{2\chi }{r^2}}\left(\begin{array}{l}2{\delta }_{kl}{r}_{,i}{r}_{,j}+{\delta }_{il}{r}_{,j}{r}_{,k}+{\delta }_{jl}{r}_{,i}{r}_{,k}+{\delta }_{jk}{r}_{,i}{r}_{,l}+{\delta }_{ik}{r}_{,j}{r}_{,l}+\\ 2{\delta }_{ij}{r}_{,l}{r}_{,k}-{\delta }_{kl}{\delta }_{ij}-6r_{,i}{r}_{,j}{r}_{,k}{r}_{,l}\end{array}\right)\\ -S_1\left(4r_{,i}{r}_{,j}{r}_{,k}{r}_{,l}+2\left({\displaystyle \frac{c_1^2}{c_2^2}}-2\right){\delta }_{kl}{r}_{,i}{r}_{,j}\right)\\ +{\displaystyle \frac{2}{r}}\left({\displaystyle \frac{c_1^2}{c_2^2}}-2\right)\left({\psi }_{,r}-{\chi }_{,r}-2{\displaystyle \frac{\chi }{r}}\right)\left({\delta }_{kl}{r}_{,i}{r}_{,j}-{\delta }_{kl}{\delta }_{ij}\right)\\ +{\displaystyle \frac{2}{r}}\left({\psi }_{,r}-{\chi }_{,r}-2{\displaystyle \frac{\chi }{r}}\right)\left(\begin{array}{l}\left({\displaystyle \frac{c_1^2}{c_2^2}}-2\right){\delta }_{ij}{r}_{,l}{r}_{,k}-\left({\displaystyle \frac{c_1^2}{c_2^2}}-2\right)\\ \left({\displaystyle \frac{c_1^2}{c_2^2}}-1\right){\delta }_{kl}{\delta }_{ij}\end{array}\right)\\ +S_1\left(2\left({\displaystyle \frac{c_1^2}{c_2^2}}-2\right){\delta }_{ij}{r}_{,l}{r}_{,k}-{\left({\displaystyle \frac{c_1^2}{c_2^2}}-2\right)}^2{\delta }_{kl}{\delta }_{ij}\right)\end{array}\right]$$

(22)

In Equation (22), S₁ and S₂ are represented as:

$$\left\{\begin{array}{l}{S}_1={\displaystyle \frac{c_1^2}{c_2^2}}{\displaystyle \frac{1}{r}}\left({\displaystyle \frac{2}{r^2}}\delta \left(t^{\text{'}}-{\displaystyle \frac{r}{c_1}}\right)+{\displaystyle \frac{2}{r}}{\displaystyle \frac{1}{c_1}}\dot{\delta }\left(t^{\text{'}}-{\displaystyle \frac{r}{c_1}}\right)+{\displaystyle \frac{1}{c_1^2}}\ddot{\delta }\left(t^{\text{'}}-{\displaystyle \frac{r}{c_1}}\right)\right)\\ S_2={\displaystyle \frac{c_1^2}{c_2^2}}{\displaystyle \frac{1}{r}}\left({\displaystyle \frac{2}{r^2}}\delta \left(t^{\text{'}}-{\displaystyle \frac{r}{c_2}}\right)+{\displaystyle \frac{2}{r}}{\displaystyle \frac{1}{c_2}}\dot{\delta }\left(t^{\text{'}}-{\displaystyle \frac{r}{c_2}}\right)+{\displaystyle \frac{1}{c_2^2}}\ddot{\delta }\left(t^{\text{'}}-{\displaystyle \frac{r}{c_2}}\right)\right)\end{array}\right.$$

(23)

It is important to note that both the displacement boundary integral equation and the stress boundary integral equation are applicable to both finite and infinite problems at the interior point [10]. The volume integral in Equation (18) exhibits a strong singularity, making it unsuitable for direct calculation of stress at the boundary point. Therefore, it is necessary to rely on Hooke’s law and boundary conditions to determine the stress equation at the boundary point. In addition, only the plastic domain needs to calculate the volume integral, rather than the entire domain. In practical engineering, the plastic domain is generally small, so the computational cost caused by volume integral is very limited, especially for infinite or semi-infinite domain problems.

The development of these kernel functions is a key contribution of this work. By providing explicit expressions for the influence coefficients, this study advances the numerical implementation of the boundary integral equations, which is particularly important for large-scale engineering problems.

4. Degradation of Fundamental Solution and Kernel Functions

4.1. Degradation of Displacement Boundary Integral Equation

In the evolution of BEM, it is a mature method to apply the fundamental solution of a 3-D plane problem to a 2-D plane problem (Niwa and Manolis) [16,17]. In view of the mature system and fundamental solution of the two-dimensional planar strain problem in TD-BEM, we can verify the correctness of the derivation of the three-dimensional fundamental solution and kernel function through the form of integral degradation.

This approach leverages the maturity of 2-D elastoplastic theory as a benchmark, circumventing the need for prohibitively complex 3-D experimental validations. The rationale for using dimensional reduction as a validation tool is twofold. Firstly, it provides a rigorous mathematical framework for verifying the correctness of the 3-D equations. By systematically reducing the dimensionality of the problem, it is possible to compare the resulting equations with well-established 2-D formulations. Secondly, it offers a practical approach to validating the 3-D equations without the need for extensive experimental data. This is particularly important in the early stages of theoretical development, where experimental validation may be challenging or impractical.

In cases where the integral degenerates, one must account for the fact that, in plane strain scenarios, such as an infinitely long cylinder along x₃ direction with loads and constrains only in plane x₁ − x₂. For the convenience of validation with dimensional reduction, Equation (1) is rewritten as follows:

$$\begin{array}{ll}{c}_{ik}{u}_i\left(P,t\right)=& -{\displaystyle {\int }_0^t{\displaystyle {\int }_{\Gamma }{\displaystyle {\int }_{-\infty }^{+\infty }{p}_{ik}^{*}{u}_k\mathrm{d}{x}_3\mathrm{d}\Gamma \mathrm{d}\tau }}}+{\displaystyle {\int }_0^t{\displaystyle {\int }_{\Gamma }{\displaystyle {\int }_{-\infty }^{+\infty }{u}_{ik}^{*}{p}_k\mathrm{d}{x}_3}}}\mathrm{d}\Gamma \mathrm{d}\tau \hfill \\ & +{\displaystyle {\int }_0^t{\displaystyle {\int }_{\Omega }{\displaystyle {\int }_{-\infty }^{+\infty }{\sigma }_{ikl}^{*}{\epsilon }_{kl}^{\mathrm{p}}\mathrm{d}{x}_3\mathrm{d}\Omega \mathrm{d}\tau }}}\end{array}$$

(24)

where ${\displaystyle {\int }_{\Gamma }{\displaystyle {\int }_{-\infty }^{+\infty }\mathrm{d}{x}_3\mathrm{d}\Gamma }}={\displaystyle {\int }_S\mathrm{d}S}$ and ${\displaystyle {\int }_{\Omega }{\displaystyle {\int }_{-\infty }^{+\infty }\mathrm{d}{x}_3\mathrm{d}\Omega }}={\displaystyle {\int }_V\mathrm{d}V}$. The integral over the surface domain S is decomposed into the integral of the boundary curve Γ perpendicular to the x₃ axis and the integral over the entire x₃ axis, and similarly, the integral over the volume domain V is decomposed into the integral over the region Ω perpendicular to the x₃ axis and the integral over the entire x₃ axis. The total strain ${\epsilon }_{33}$ in the inner field, the traction p₃, and displacement u₃ on the boundary are zero. However, the stress σ₃₃ is not zero, which indicates that the corresponding elastic strain ${\epsilon }_{33}^{\mathrm{e}}$ is not zero. That is, the plastic strain ${\epsilon }_{33}^{\mathrm{p}}$ is not zero, as shown in Equation (25). Since the component ${\sigma }_{33}^{*}$ is not zero, the integral related to ${\epsilon }_{33}^{\mathrm{p}}$ remains, while the integral related to p₃ and u₃ disappears. Additionally, the plastic shear strain components ${\epsilon }_{13}^{\mathrm{p}}$, ${\epsilon }_{23}^{\mathrm{p}}$, ${\epsilon }_{31}^{\mathrm{p}}$, and ${\epsilon }_{32}^{\mathrm{p}}$ are all zero. Then, Equation (24) becomes Equation (26).

$${\epsilon }_{33}^{\mathrm{p}}={\epsilon }_{33}-{\epsilon }_{33}^{\mathrm{e}}=-{\epsilon }_{33}^{\mathrm{e}}$$

(25)

$$\begin{array}{ll}{c}_{ik}{u}_i\left(P,t\right)=& -{\displaystyle {\int }_0^t{\displaystyle {\int }_{\Gamma }{\displaystyle {\int }_{-\infty }^{+\infty }{p}_{ik}^{*}d{x}_3{u}_k\mathrm{d}\Gamma \mathrm{d}\tau }}}+{\displaystyle {\int }_0^t{\displaystyle {\int }_{\Gamma }{\displaystyle {\int }_{-\infty }^{+\infty }{u}_{ik}^{*}\mathrm{d}{x}_3}}}{p}_k\mathrm{d}\Gamma \mathrm{d}\tau \hfill \\ & +{\displaystyle {\int }_0^t{\displaystyle {\int }_{\Omega }{\displaystyle {\int }_{-\infty }^{+\infty }\left({\sigma }_{ikl}^{*}{\epsilon }_{kl}^{\mathrm{p}}+{\sigma }_{i33}^{*}{\epsilon }_{33}^{\mathrm{p}}\right)\mathrm{d}{x}_3\mathrm{d}\Omega \mathrm{d}\tau }}}\end{array}$$

(26)

where i, k, l = 1, 2. And the values of i, k, l are the same as in the following parts of Section 4.

In order to facilitate the transformation of the integral related to ${\epsilon }_{33}^{\mathrm{p}}$, the assumption of zero plastic volumetric strain is introduced, as shown in Equations (27) and (28).

$${\epsilon }^{\mathrm{p}}={\epsilon }_{33}^{\mathrm{p}}+{\epsilon }_{11}^{\mathrm{p}}+{\epsilon }_{22}^{\mathrm{p}}=0$$

(27)

$${\epsilon }_{33}^{\mathrm{p}}=-\left({\epsilon }_{11}^{\mathrm{p}}+{\epsilon }_{22}^{\mathrm{p}}\right)=-{\epsilon }_{kl}^{\mathrm{p}}{\delta }_{kl}$$

(28)

Substituting Equation (28) into the third integral of Equation (26), one has:

$${\displaystyle {\int }_0^t{\displaystyle {\int }_{\Omega }{\displaystyle {\int }_{-\infty }^{+\infty }\left({\sigma }_{ikl}^{*}{\epsilon }_{kl}^{\mathrm{p}}+{\sigma }_{i33}^{*}{\epsilon }_{33}^{\mathrm{p}}\right)\mathrm{d}{x}_3\mathrm{d}\Omega \mathrm{d}\tau }}}={\displaystyle {\int }_0^t{\displaystyle {\int }_{\Omega }{\displaystyle {\int }_{-\infty }^{+\infty }{\overline{\sigma }}_{ikl}^{*}\mathrm{d}{x}_3{\epsilon }_{kl}^{\mathrm{p}}\mathrm{d}\Omega \mathrm{d}\tau }}}$$

(29)

where ${\overline{\sigma }}_{ikl}^{*}$ is equivalent stress fundamental solution, which is expressed as:

$${\overline{\sigma }}_{ikl}^{*}={\sigma }_{ikl}^{*}-{\sigma }_{i33}^{*}{\delta }_{kl}$$

(30)

Further clarification is required to emphasize that, in contrast to 3-D problems, ${\overline{\sigma }}_{ijk}^{*}$ serves as the equivalent stress fundamental solution in the context of plane strain problems rather than plane stress. Given the potential for plastic strain in the third direction in plane strain problems, we need to calculate the influence of ${\epsilon }_{33}^{\mathrm{p}}$ by applying what is known as equivalent stress fundamental solution ${\overline{\sigma }}_{ijk}^{*}$. Therefore, Equation (26) can be rewritten as:

$$\begin{array}{ll}{c}_{ik}{u}_i\left(P,t\right)=& -{\displaystyle {\int }_0^t{\displaystyle {\int }_{\Gamma }{\displaystyle {\int }_{-\infty }^{+\infty }{p}_{ik}^{*}d{x}_3{u}_k\mathrm{d}\Gamma \mathrm{d}\tau }}}+{\displaystyle {\int }_0^t{\displaystyle {\int }_{\Gamma }{\displaystyle {\int }_{-\infty }^{+\infty }{u}_{ik}^{*}\mathrm{d}{x}_3}}}{p}_k\mathrm{d}\Gamma \mathrm{d}\tau \hfill \\ & +{\displaystyle {\int }_0^t{\displaystyle {\int }_{\Omega }{\displaystyle {\int }_{-\infty }^{+\infty }{\overline{\sigma }}_{ikl}^{*}\mathrm{d}{x}_3{\epsilon }_{kl}^{\mathrm{p}}\mathrm{d}\Omega \mathrm{d}\tau }}}\end{array}$$

(31)

Equation (31) is the degradation form of Equation (1) based on the plane strain problem.

4.2. Degradation of Stress Boundary Integral Equation

Similar processes are applied to the stress boundary integral Equation (18) where Equation (32) is obtained, which is the degradation form of Equation (18) based on the plane strain problem.

$$\begin{array}{ll}{\sigma }_{ij}\left(P,t\right)=& -{\displaystyle {\int }_0^t{\displaystyle {\int }_{\Gamma }{\displaystyle {\int }_{-\infty }^{+\infty }{s}_{ijk}^{*}\mathrm{d}{x}_3{u}_k\mathrm{d}\Gamma \mathrm{d}\tau }}}+{\displaystyle {\int }_0^t{\displaystyle {\int }_{\Gamma }{\displaystyle {\int }_{-\infty }^{+\infty }{d}_{ijk}^{*}\mathrm{d}{x}_3{p}_k\mathrm{d}\Gamma \mathrm{d}\tau }}}\hfill \\ & +{\displaystyle {\int }_0^t{\displaystyle {\int }_{\Omega }{\displaystyle {\int }_{-\infty }^{+\infty }{\overline{\sigma }}_{ijkl}^{*}\mathrm{d}{x}_3{\epsilon }_{kl}^{\mathrm{p}}\mathrm{d}\Omega \mathrm{d}\tau }}}-{\sigma }_{ij}^{\mathrm{p}}\end{array}$$

(32)

4.3. Degradation of Fundamental Solutions and Kernel Functions

By comparing the transformed boundary integral equations in Equations (31) and (32) with the boundary integral equations of the plane strain problem in reference [14], it can be seen that the 2D fundamental solutions can be obtained by integrating the 3D fundamental solutions along the x₃ axis. The relationship between them is shown in Equation (33).

$$U^{2D}={\displaystyle {\int }_{-\infty }^{+\infty }{U}^{3D}}d{x}_3$$

(33)

The time-domain fundamental solution of displacement, traction, and stress for 2-D plane strain problems are obtained as:

$$u_{ik}^{*}(P,\tau ;Q,t)={\displaystyle \frac{1}{2\pi \rho }}\left\{{\displaystyle \frac{1}{c_1}}{H}_1\left[B_1{C}_1^{-{\displaystyle \frac{1}{2}}}{\displaystyle \frac{r_{,i}{r}_{,k}}{r}}-{\displaystyle \frac{{\delta }_{ik}}{r}}{C}_1^{{\displaystyle \frac{1}{2}}}\right]+{\displaystyle \frac{1}{c_2}}{H}_2\left[{\displaystyle \frac{{\delta }_{ik}}{r}}{\displaystyle \frac{B_2+1}{2}}{C}_2^{-{\displaystyle \frac{1}{2}}}-B_2{C}_2^{-{\displaystyle \frac{1}{2}}}{\displaystyle \frac{r_{,i}{r}_{,k}}{r}}\right]\right\}$$

(34)

$$p_{ik}^{*}(P,\tau ;Q,t)={\displaystyle \frac{\mu }{2\pi \rho r}}\left\{{\displaystyle \frac{1}{c_1}}{H}_1\left[C_1^{-{\displaystyle \frac{3}{2}}}{\displaystyle \frac{D_1}{r}}+\left(2A_1+1\right)C_1^{-{\displaystyle \frac{1}{2}}}{\displaystyle \frac{2D_2}{r}}\right]-{\displaystyle \frac{1}{c_2}}{H}_2\left[C_2^{-{\displaystyle \frac{3}{2}}}{\displaystyle \frac{D_3}{r}}+B_2{C}_2^{-{\displaystyle \frac{1}{2}}}{\displaystyle \frac{2D_2}{r}}\right]\right\}$$

(35)

$${\overline{\sigma }}_{ikl}^{*}(P,\tau ;Q,t)={\displaystyle \frac{\mu }{2\pi \rho r}}\left\{{\displaystyle \frac{1}{c_1}}{H}_1\left[C_1^{-{\displaystyle \frac{3}{2}}}{\displaystyle \frac{E_1}{r}}+\left(2A_1+1\right)C_1^{-{\displaystyle \frac{1}{2}}}{\displaystyle \frac{2E_2}{r}}\right]-{\displaystyle \frac{1}{c_2}}{H}_2\left[C_2^{-{\displaystyle \frac{3}{2}}}{\displaystyle \frac{E_3}{r}}+B_2{C}_2^{-{\displaystyle \frac{1}{2}}}{\displaystyle \frac{2E_2}{r}}\right]\right\}$$

(36)

Further combined with Equation (19), the 2-D time-domain influence coefficient kernel function is obtained as $s_{ijk}^{*}$, $d_{ijk}^{*}$ and ${\overline{\sigma }}_{ijkl}^{*}$, respectively, which are expressed as:

$$d_{ijk}^{*}(P,\tau ;Q,t)={\displaystyle \frac{\mu }{2\pi \rho r}}\left\{{\displaystyle \frac{1}{c_2}}{H}_2\left[C_2^{-{\displaystyle \frac{3}{2}}}{\displaystyle \frac{E_3}{r}}+B_2{C}_2^{-{\displaystyle \frac{1}{2}}}{\displaystyle \frac{2E_2}{r}}\right]-{\displaystyle \frac{1}{c_1}}{H}_1\left[C_1^{-{\displaystyle \frac{3}{2}}}{\displaystyle \frac{E_1}{r}}+B_1{C}_1^{-{\displaystyle \frac{1}{2}}}{\displaystyle \frac{2E_2}{r}}\right]\right\}$$

(37)

$$\begin{array}{l}{s}_{ijk}^{*}(P,\tau ;Q,t)={\displaystyle \frac{{\mu }^2}{2\pi \rho r^2}}\left\{{\displaystyle \frac{1}{c_2}}{H}_2\left[C_2^{-{\displaystyle \frac{5}{2}}}{\displaystyle \frac{3F_1}{r}}-C_2^{-{\displaystyle \frac{3}{2}}}{\displaystyle \frac{2F_2}{r}}-B_2{C}_2^{-{\displaystyle \frac{1}{2}}}{\displaystyle \frac{4F_3}{r}}\right]\right.\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.33em}\hspace{1em}\left.-{\displaystyle \frac{1}{c_1}}{H}_1\left[C_1^{-{\displaystyle \frac{5}{2}}}{\displaystyle \frac{3F_4}{r}}-C_1^{-{\displaystyle \frac{3}{2}}}{\displaystyle \frac{2F_5}{r}}-B_1{C}_1^{-{\displaystyle \frac{1}{2}}}{\displaystyle \frac{4F_3}{r}}\right]\right\}\end{array}$$

(38)

$$\begin{array}{l}{\overline{\sigma }}_{ijkl}^{*}(P,\tau ;Q,t)={\displaystyle \frac{{\mu }^2}{2\pi \rho r^2}}\left\{{\displaystyle \frac{1}{c_2}}{H}_2\left[C_2^{-{\displaystyle \frac{5}{2}}}{\displaystyle \frac{3G_1}{r}}-C_2^{-{\displaystyle \frac{3}{2}}}{\displaystyle \frac{2G_2}{r}}-B_2{C}_2^{-{\displaystyle \frac{1}{2}}}{\displaystyle \frac{4G_3}{r}}\right]\right.\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.33em}\hspace{1em}\left.-{\displaystyle \frac{1}{c_1}}{H}_1\left[C_1^{-{\displaystyle \frac{5}{2}}}{\displaystyle \frac{3G_4}{r}}-C_1^{-{\displaystyle \frac{3}{2}}}{\displaystyle \frac{2G_5}{r}}-B_1{C}_1^{-{\displaystyle \frac{1}{2}}}{\displaystyle \frac{4G_3}{r}}\right]\right\}\end{array}$$

(39)

The correlation coefficients in fundamental solutions of the 2-D plane strain problem above are expressed as follows:

$$\begin{array}{l}{H}_w=H\left(A_w\right)\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}w=1,2\\ A_w={\displaystyle \frac{c_w{t}^{\prime }}{r}}-1\\ B_w=2{\left({\displaystyle \frac{c_w{t}^{\prime }}{r}}\right)}^2-1\\ C_w={\left({\displaystyle \frac{c_w{t}^{\prime }}{r}}\right)}^2-1\end{array}$$

$$\begin{array}{l}{D}_1={\displaystyle \frac{\lambda }{\mu }}{n}_i{r}_{,j}+2r_{,i}{r}_{,j}{\displaystyle \frac{\partial r}{\partial n}}\\ D_2=n_i{r}_{,j}+n_j{r}_{,i}+{\displaystyle \frac{\partial r}{\partial n}}\left({\delta }_{ij}-4r_{,i}{r}_{,j}\right)\\ D_3={\displaystyle \frac{\partial r}{\partial n}}\left(2r_{,i}{r}_{,j}-{\delta }_{ij}\right)-n_j{r}_{,i}\end{array}$$

$$\begin{array}{l}{E}_1={\displaystyle \frac{\lambda }{\mu }}{\delta }_{ik}{r}_{,j}+2r_{,i}{r}_{,j}{r}_{,k}\\ E_2={\delta }_{ik}{r}_{,j}+{\delta }_{jk}{r}_{,i}+r_{,k}\left({\delta }_{ij}-4r_{,i}{r}_{,j}\right)\\ E_3=r_{,k}\left(2r_{,i}{r}_{,j}-{\delta }_{ij}\right)-{\delta }_{jk}{r}_{,i}\end{array}$$

$$\begin{array}{l}{F}_1=r_{,i}{n}_i\left(4r_{,i}{r}_{,j}{r}_{,k}-r_{,i}{\delta }_{jk}-r_{,j}{\delta }_{ik}\right)-\left(n_i{r}_{,j}{r}_{,k}+n_j{r}_{,i}{r}_{,k}\right)\\ F_2=n_i\left({\delta }_{jk}-2r_{,j}{r}_{,k}\right)+n_j\left({\delta }_{ik}-2r_{,i}{r}_{,k}\right)-2n_k{r}_{,i}{r}_{,j}\\ -2r_{,i}{n}_i\left(r_{,i}{\delta }_{jk}+r_{,j}{\delta }_{ik}+r_{,k}{\delta }_{ij}-6r_{,i}{r}_{,j}{r}_{,k}\right)\\ F_3=n_i\left(4r_{,j}{r}_{,k}-{\delta }_{jk}\right)+n_j\left(4r_{,i}{r}_{,k}-{\delta }_{ik}\right)+n_k\left(4r_{,i}{r}_{,j}-{\delta }_{ij}\right)\\ +4r_{,i}{n}_i\left(r_{,i}{\delta }_{jk}+r_{,j}{\delta }_{ik}+r_{,k}{\delta }_{ij}-6r_{,i}{r}_{,j}{r}_{,k}\right)\\ F_4=\left({\displaystyle \frac{\lambda }{\mu }}{\delta }_{ij}+2r_{,i}{r}_{,j}\right)\left({\displaystyle \frac{\lambda }{\mu }}{n}_k+2r_{,i}{n}_i{r}_{,k}\right)\\ F_5=n_k\left[-{\displaystyle \frac{2\lambda }{\mu }}{\delta }_{ij}-{\left({\displaystyle \frac{\lambda }{\mu }}\right)}^2{\delta }_{ij}-2r_{,i}{r}_{,j}\right]-2\left(n_i{r}_{,j}{r}_{,k}+n_j{r}_{,i}{r}_{,k}\right)\\ -2r_{,i}{n}_i\left(r_{,i}{\delta }_{jk}+r_{,j}{\delta }_{ik}+r_{,k}{\delta }_{ij}-6r_{,i}{r}_{,j}{r}_{,k}\right)\end{array}$$

In reference [14], Li et al. employed the initial strain method to establish the boundary integral equation for the 2-D dynamic elastoplastic TD-BEM plane problem and derived explicit expressions for the time-domain fundamental solution, including displacement, traction, equivalent stress, and their corresponding influence coefficient kernel functions. This study utilized the initial strain method to integrate the 3-D time-domain fundamental solution into its corresponding 2-D plane strain problem. Comparing this with Li et al.’s 2-D formulation revealed strict equivalence, validating both the mathematical degradation and the physical consistency of the initial strain method.

After being organized into the same form as references [9,14], Equations (34)–(36) are rewritten as:

$$u_{ik}^{*}={\displaystyle \frac{1}{2\pi \rho c_s}}\left[\left(E_{ik}{L}_s+F_{ik}{L}_s^{-1}+J_{ik}{L}_s{N}_s\right)H_s-{\displaystyle \frac{c_s}{c_d}}\left(F_{ik}{L}_d^{-1}+J_{ik}{L}_d{N}_d\right)H_d\right]$$

(40)

$$\begin{array}{ll}{p}_{ik}^{*}=& {\displaystyle \frac{1}{2\pi \rho c_s}}\left\{A_{ik}\left(r{L}_s^3{H}_s+L_s{\displaystyle \frac{\partial H_s}{\partial \left(c_s\tau \right)}}\right)+B_{ik}{L}_s{N}_s{H}_s+{\displaystyle \frac{D_{ik}}{r^2}}\left(r^3{L}_s^3{H}_s+L_s{N}_s{\displaystyle \frac{\partial H_s}{\partial \left(c_s\tau \right)}}\right)\right.\hfill \\ & \left.-{\displaystyle \frac{c_s}{c_d}}\left[B_{ik}{L}_d{N}_d{H}_d+{\displaystyle \frac{D_{ik}}{r^2}}\left(r^3{L}_d^3{H}_d+L_d{N}_d{\displaystyle \frac{\partial H_d}{\partial \left(c_d\tau \right)}}\right)\right]\right\}\end{array}$$

(41)

$$\begin{array}{ll}{\overline{\sigma }}_{ikl}^{*}=& {\displaystyle \frac{1}{2\pi \rho c_s}}\left\{A_{ikl}\left(r{L}_s^3{H}_s+L_s{\displaystyle \frac{\partial H_s}{\partial \left(c_s\tau \right)}}\right)+B_{ikl}{L}_s{N}_s{H}_s+{\displaystyle \frac{D_{ikl}}{r^2}}\left(r^3{L}_s^3{H}_s+L_s{N}_s{\displaystyle \frac{\partial H_s}{\partial \left(c_s\tau \right)}}\right)\right.\hfill \\ & \left.-{\displaystyle \frac{c_s}{c_d}}\left[B_{ikl}{L}_d{N}_d{H}_d+{\displaystyle \frac{D_{ikl}}{r^2}}\left(r^3{L}_d^3{H}_d+L_d{N}_d{\displaystyle \frac{\partial H_d}{\partial \left(c_d\tau \right)}}\right)\right]\right\}\end{array}$$

(42)

The kernel functions in Equations (37)–(39) are rewritten as:

$$d_{ijk}^{*}(P,\tau ;Q,t)={\displaystyle \frac{1}{2\pi \rho c_s^2}}\left[\begin{array}{l}\left(\begin{array}{l}{E}_{ijk}{c}_s{L}_s{H}_s+J_{ijk}^0{c}_d{\displaystyle \frac{\partial \left(L_d{N}_d\right)}{\partial r}}{H}_d\\ +F_{ijk}{c}_s{L}_s^{-1}{H}_s+F_{ijk}^0{c}_d{\displaystyle \frac{\partial \left(L_s^{-1}\right)}{\partial r}}{H}_s\\ +J_{ijk}{c}_s{L}_s{N}_s{H}_s+F_{ijk}^0{c}_d{\displaystyle \frac{\partial \left(L_s^{-1}\right)}{\partial r}}{H}_s\end{array}\right)\\ -{\displaystyle \frac{c_s^2}{c_d^2}}\left(\begin{array}{l}{F}_{ijk}{c}_d{L}_d^{-1}{H}_d+F_{ijk}^0{c}_d{\displaystyle \frac{\partial \left(L_d^{-1}\right)}{\partial r}}{H}_d\\ +J_{ijk}{c}_d{L}_d{N}_d{H}_d+J_{ijk}^0{c}_d{\displaystyle \frac{\partial \left(L_d{N}_d\right)}{\partial r}}{H}_d\end{array}\right)\end{array}\right]$$

(43)

$$s_{ijk}^{*}(P,\tau ;Q,t)={\displaystyle \frac{1}{2\pi \rho c_s^2}}\left[\begin{array}{l}\left(\begin{array}{l}\left(A_{ijk}+D_{ijk}\right)c_{s}r{L}_s^3{H}_s+\left(A_{ijk}^0+D_{ijk}^0\right){\displaystyle \frac{\partial }{\partial r}}{c}_{s}r{L}_s^3{H}_s\\ +B_{ijk}{c}_s{L}_s{N}_s{H}_s+B_{ijk}^0{c}_s{\displaystyle \frac{\partial \left(L_s{N}_s\right)}{\partial r}}{H}_s\end{array}\right)\\ -{\displaystyle \frac{c_s^2}{c_d^2}}\left(\begin{array}{l}{B}_{ijk}{c}_d{L}_d{N}_d{H}_d+B_{ijk}^0{c}_d{\displaystyle \frac{\partial \left(L_d{N}_d\right)}{\partial r}}{H}_d\\ +D_{ijk}{c}_{d}r{L}_d^3{H}_d+D_{ijk}^0{\displaystyle \frac{\partial }{\partial r}}{c}_{d}r{L}_d^3{H}_d\end{array}\right)\end{array}\right]$$

(44)

$${\overline{\sigma }}_{ijkl}^{*}(P,\tau ;Q,t)={\displaystyle \frac{1}{2\pi \rho c_s^2}}\left[\begin{array}{l}\left(\begin{array}{l}\left(A_{ijkl}+D_{ijkl}\right)c_{s}r{L}_s^3{H}_s+\left(A_{ijkl}^0+D_{ijkl}^0\right){\displaystyle \frac{\partial }{\partial r}}{c}_{s}r{L}_s^3{H}_s\\ +B_{ijkl}{c}_s{L}_s{N}_s{H}_s+B_{ijkl}^0{c}_s{\displaystyle \frac{\partial \left(L_s{N}_s\right)}{\partial r}}{H}_s\end{array}\right)\\ -{\displaystyle \frac{c_s^2}{c_d^2}}\left(\begin{array}{l}{B}_{ijkl}{c}_d{L}_d{N}_d{H}_d+B_{ijkl}^0{c}_d{\displaystyle \frac{\partial \left(L_d{N}_d\right)}{\partial r}}{H}_d\\ +D_{ijkl}{c}_{d}r{L}_d^3{H}_d+D_{ijkl}^0{\displaystyle \frac{\partial }{\partial r}}{c}_{d}r{L}_d^3{H}_d\end{array}\right)\end{array}\right]$$

(45)

According to references [10,14], Equation (32) can be organized into the form of surface traction integral term, displacement integral term, and plastic strain integral term:

$${\sigma }_{ij}\left(P,t\right)={\displaystyle \frac{1}{2\pi \rho c_s^2}}\left(D_{sd}^p-S_{sd}^u-{\Sigma }_{sd}^{\epsilon }\right)+f_{ij}^P$$

(46)

Equations (43)–(45) are included in the above integral term, and the specific expressions of the integral terms are:

$$\begin{array}{l}{D}_{sd}^p={\displaystyle {\int }_0^t{\displaystyle {\int }_{\Gamma }{d}_{ijk}^{*}{p}_k\mathrm{d}\Gamma \mathrm{d}\tau }}\\ \hspace{1em}\hspace{0.33em}={\displaystyle {\int }_{\Gamma }\left\{\begin{array}{l}\left[\begin{array}{l}{E}_{ijk}{\displaystyle {\int }_0^t{c}_s{L}_s{p}_k{H}_{s}d\tau }+E_{ijk}^0\overline{){\displaystyle {\int }_0^t{c}_s\frac{\partial L_s}{\partial r}{p}_k{H}_{s}d\tau }}\\ +F_{ijk}{\displaystyle {\int }_0^t{c}_s{L}_s^{-1}{p}_k{H}_{s}d\tau }+F_{ijk}^0\overline{){\displaystyle {\int }_0^t{c}_s\frac{\partial \left(L_s^{-1}\right)}{\partial r}{p}_k{H}_{s}d\tau }}\\ +J_{ijk}{\displaystyle {\int }_0^t{c}_s{L}_s{N}_s{p}_k{H}_{s}d\tau }+J_{ijk}^0\overline{){\displaystyle {\int }_0^t{c}_s\frac{\partial \left(L_s{N}_s\right)}{\partial r}{p}_k{H}_{s}d\tau }}\end{array}\right]\\ -\frac{c_s^2}{c_d^2}\left[\begin{array}{l}{F}_{ijk}{\displaystyle {\int }_0^t{c}_d{L}_d^{-1}{p}_k{H}_{d}d\tau }+F_{ijk}^0\overline{){\displaystyle {\int }_0^t{c}_d\frac{\partial \left(L_d^{-1}\right)}{\partial r}{p}_k{H}_{d}d\tau }}\\ +J_{ijk}{\displaystyle {\int }_0^t{c}_d{L}_d{N}_d{p}_k{H}_{d}d\tau }+J_{ijk}^0\overline{){\displaystyle {\int }_0^t{c}_d\frac{\partial \left(L_d{N}_d\right)}{\partial r}{p}_k{H}_{d}d\tau }}\end{array}\right]\end{array}\right\}}d\Gamma \end{array}$$

(47)

$$\begin{array}{l}{S}_{sd}^u={\displaystyle {\int }_0^t{\displaystyle {\int }_{\Gamma }{s}_{ijk}^{*}{u}_k\mathrm{d}\Gamma \mathrm{d}\tau }}\\ \hspace{1em}\hspace{0.33em}={\displaystyle {\int }_{\Gamma }\left\{\begin{array}{l}\left[\begin{array}{l}\left(A_{ijk}+D_{ijk}\right)\overline{){\displaystyle {\int }_0^t{c}_{s}r{L}_s^3{u}_k{H}_{s}d\tau }}+\left(A_{ijk}^0+D_{ijk}^0\right)\frac{\partial }{\partial r}\overline{){\displaystyle {\int }_0^t{c}_{s}r{L}_s^3{u}_k{H}_{s}d\tau }}\\ +B_{ijk}{\displaystyle {\int }_0^t{c}_s{L}_s{N}_s{u}_k{H}_{s}d\tau }+B_{ijk}^0\overline{){\displaystyle {\int }_0^t{c}_s\frac{\partial \left(L_s{N}_s\right)}{\partial r}{u}_k{H}_{s}d\tau }}\end{array}\right]\\ -\frac{c_s^2}{c_d^2}\left[\begin{array}{l}{B}_{ijk}{\displaystyle {\int }_0^t{c}_d{L}_d{N}_d{u}_k{H}_{d}d\tau }+B_{ijk}^0\overline{){\displaystyle {\int }_0^t{c}_d\frac{\partial \left(L_d{N}_d\right)}{\partial r}{u}_k{H}_{d}d\tau }}\\ +D_{ijk}\overline{){\displaystyle {\int }_0^t{c}_{d}r{L}_d^3{u}_k{H}_{d}d\tau }}+D_{ijk}^0\frac{\partial }{\partial r}\overline{){\displaystyle {\int }_0^t{c}_{d}r{L}_d^3{u}_k{H}_{d}d\tau }}\end{array}\right]\end{array}\right\}}d\Gamma \end{array}$$

(48)

$$\begin{array}{l}{\Sigma }_{sd}^{\epsilon }={\displaystyle {\int }_0^t{\displaystyle {\int }_{\Gamma }{s}_{ijk}^{*}{u}_k\mathrm{d}\Gamma \mathrm{d}\tau }}\\ \hspace{1em}\hspace{0.33em}={\displaystyle {\int }_{\Omega }\left\{\begin{array}{l}\left[\begin{array}{l}\left(A_{ijkl}+D_{ijkl}\right)\overline{){\displaystyle {\int }_0^t{c}_{s}r{L}_s^3{\epsilon }_{kl}^p{H}_{s}d\tau }}+\left(A_{ijkl}^0+D_{ijkl}^0\right)\frac{\partial }{\partial r}\overline{){\displaystyle {\int }_0^t{c}_{s}r{L}_s^3{\epsilon }_{kl}^p{H}_{s}d\tau }}\\ +B_{ijkl}{\displaystyle {\int }_0^t{c}_s{L}_s{N}_s{\epsilon }_{kl}^p{H}_{s}d\tau }+B_{ijkl}^0\overline{){\displaystyle {\int }_0^t{c}_s\frac{\partial \left(L_s{N}_s\right)}{\partial r}{\epsilon }_{kl}^p{H}_{s}d\tau }}\end{array}\right]\\ -\frac{c_s^2}{c_d^2}\left[\begin{array}{l}{B}_{ijkl}{\displaystyle {\int }_0^t{c}_d{L}_d{N}_d{\epsilon }_{kl}^p{H}_{d}d\tau }+B_{ijkl}^0\overline{){\displaystyle {\int }_0^t{c}_d\frac{\partial \left(L_d{N}_d\right)}{\partial r}{\epsilon }_{kl}^p{H}_{d}d\tau }}\\ +D_{ijkl}{\displaystyle {\int }_0^t{c}_{d}r{L}_d^3{\epsilon }_{kl}^p{H}_{d}d\tau }+D_{ijkl}^0\frac{\partial }{\partial r}\overline{){\displaystyle {\int }_0^t{c}_{d}r{L}_d^3{\epsilon }_{kl}^p{H}_{d}d\tau }}\end{array}\right]\end{array}\right\}}d\Omega \end{array}$$

(49)

The validation of integral degradation is a key aspect of this work. By demonstrating that the 3-D equations are reduced to well-established 2-D formulations, the study provides evidence for the correctness of the proposed method.

5. Conclusions

The primary objective of this study was to establish a theoretical framework for three-dimensional (3-D) dynamic elastoplastic analysis by using the time-domain boundary element method (TD-BEM) based on the initial strain method. The key findings and contributions of this work can be summarized as follows:

• For the first time, a complete set of 3-D dynamic elastoplastic boundary integral equations was derived using the initial strain method the unlike traditional initial stress method.
• Explicit expressions for displacement, traction, and strain influence kernel functions were derived by coupling time-domain fundamental solutions with physical and geometric equations. These kernels enable the transformation of boundary integral equations into numerically tractable forms.
• Through a systematic integral degradation process, the 3-D elastoplastic TD-BEM equations were reduced to a 2-D plane strain system. The degenerated solutions exhibited equivalence with well-established 2-D formulations in reference [12], confirming the correctness and universality of the proposed 3-D framework.

This study verified the correctness of the proposed formulas, which lays a foundation for the numerical implementation and engineering application of the TD-BEM for 3-D elastoplastic dynamics. In the process of numerical implementation, integral singularity and time-space domain division are the main factors affecting the accuracy and stability of the numerical solution of TD-BEM. In the following research, we will focus on addressing these issues to achieve efficient solutions for 3-D elastoplastic dynamics.