BEM Modeling for Stress Sensitivity of Nonlocal Thermo-Elasto-Plastic Damage Problems

: The main objective of this paper is to propose a new boundary element method (BEM) modeling for stress sensitivity of nonlocal thermo-elasto-plastic damage problems. The numerical solution of the heat conduction equation subjected to a non-local condition is described using a boundary element model. The total amount of heat energy contained inside the solid under consideration is speci ﬁ ed by the non-local condition. The procedure of solving the heat equation will reveal an unknown control function that governs the temperature on a speci ﬁ c region of the solid’s boundary. The initial stress BEM for structures with strain-softening damage is employed in a boundary element program with iterations in each load increment to develop a plasticity model with yield limit deterioration. To avoid the di ﬃ culties associated with the numerical calculation of singular integrals, the regularization technique is applicable to integral operators. To validate the physical correctness and e ﬃ ciency of the suggested formulation, a numerical case is solved.


Introduction
The non-classical heat conduction problem, which requires solving the parabolic heat equation under non-local conditions, is especially important here.A domain integral specifies the total amount of heat energy contained in the solid under consideration, resulting in the non-local state.The temperature is set in a specific section of the solid's border using an unknown control function to be determined.Noye et al. [1], Gumel et al. [2], Dehghan [3], and many other authors have numerically addressed this problem using finite difference methods.In the non-local case, the domain integral was reduced to an integral that only included the heat flow on the solution domain's border.The physical solution was obtained from the domain of the Laplace transformation using a computational technique for inverting Laplace transformation.Ang and Ooi [4] recently presented a boundary element technique for solving axisymmetric heat equation subject to nonlocal condition.
Continuum damage mechanics were created to fill the gap between classical continuum mechanics and fracture mechanics.Due to the presence of a fracture process zone with multiple microcracks at the fracture front, classical linear elastic fracture mechanics cannot be applied to heterogeneous quasi-brittle materials such as concrete, rocks, sea ice, fiber composites, and toughened ceramics unless very large structures are considered.Microcracking in materials eventually causes strain softening, which causes the elastic modulus matrix to become a non-positive definite, resulting in an ill-posed problem [5,6].Finite element simulations with elasto-plastic models and yield limit deterioration in the traditional theory of plasticity produce varied findings depending on the discretization meshes [7].The results ignore finite element mesh refinements and lead to a solution with no energy dissipation upon failure at infinite mesh refinement.To avoid this wrong behavior, localization limiters must be implemented, ensuring that the strain-softening zone has a minimum finite size [8][9][10][11][12][13].The non-local continuum idea, which was first presented in elasticity for a different reason [14,15], is an effective localization restriction.The non-local continuum with local strain, as proposed in References [16,17], is a useful model that treats just the elements that cause strain softening as non-local while treating all other variables as local.This idea is applicable to all constitutive models [13].Jirásek [18] investigated non-local models, whereas Baant and Jirásek [19] conducted considerable research.The boundary element technique (BEM) is an increasingly popular alternative to the finite element method (FEM).The key advantage over domain techniques is that there are fewer unknowns.It is a powerful linear elasticity technique that has also proven useful in inelastic material challenges.Swedlow and Cruse [20] developed BEM formulations for elasto-plasticity.Ricardella [21] pioneered the initial 2D study, whereas Chaudonneret [22] and Kumar and Mukherjee [23] reported the first 2D viscoplastic analyses.Banerjee et al. also contributed to this study [24].Banerjee and Cathie provided the first axisymmetric elasto-plastic investigations, as well as the first 3D elasto-plastic applications [25].Since then, inelastic formulations have advanced greatly [26][27][28][29].To resolve inelastic material difficulties, the BEM solution algorithm requires precise stress rate estimation.Regularization or singularity reduction prior to numerical computation can accelerate and improve the BEM solution process.
In this study, a novel boundary element method (BEM) modeling is utilized to numerically compute the temperature field in a body containing a preset amount of heat energy.A plasticity model with yield limit deterioration based on the regularized initial stress boundary element formulation is used to deal with non-local softening damage.This formulation eliminates one source of numerical error by utilizing a regularized integral representation of stresses.The BEM approach finally reduces the problem to a system of linear algebraic equations that need to be solved at each time level.Because of the regularization or singularity reduction performed before to numerical calculation, the validity, accuracy, and efficiency of the proposed BEM formulation are demonstrated.
In other words, (35) represents a system of 2 +  linear algebraic equations in 2 +  + 1 unknown functions of  if  ( )  − Δ ( = 1,2, … ,2 + ) is taken to be known.(Remember that the control function (), which is shown in (10), is an unidentified function that needs to be found).Therefore, the system requires one further equation to be completed.

BEM Modeling of Thermo-Elasto-Plastic Deformation
The integral equation for the displacement rate is as follows [35]: where the fundamental displacement  must satisfy the following equation The other kernel functions of Equation ( 39) are as follows [35]  (, ) where in which As a result, we have the following boundary integral equation (BIE) Upon substituting the displacement rate integral Equation ( 39) into (4) and applying regularization [36], we obtain the following stress rate integral equation where where () is the unit vector tangent to Γ at .
To treat a strong singularity of the kernel, the domain Ω can be split into regular and singular parts, Ω = Ω + Ω ; thus, we can write With where The BIE (53) can be used to estimate the stress tensor rates as follows By solving the regularized BIE [37], with using (51), we obtain Thus, by using (51), we have the following relations Using the subtraction and addition technique [38] and  =  ∈ Γ for the regularized BIE, we obtain where The incremental iterative solution approach will employ the regularized BIE ( 50) and (60) as well as the regularized BIE (53).Thus, the regular Gaussian quadrature rule has been used to calculate all the boundary and domain integrals.
The yield function can be expressed as For the model in this study, the work-hardening function is used to characterize softening, where the hardening/softening parameter  the plastic strain increment d can be defined as where  = ∂/ ∂ .From ( 62) and (64), the proportionality coefficient d can be expressed as We will consider the strain-softening von Mises plasticity model in this work because the nonlocal BEM solution will be easier and more obvious.
The stress intensity in which The fundamental idea behind the non-local continuum is that only the constitutive equation variables that cause strain softening are non-local [7].Spatial averaging is more computationally efficient than plastic strains Then, the non-local scaling parameter  ‾ can be defined as where where the non-local weight function (, ) can be expressed as [7] (, ) = exp(−(2/) ) where  measures the material heterogeneity scale and  = | − |.
After determining the non-local average d ‾ , we can calculate the non-local plastic strain and stress increments as follows d‾ = d ‾  (76) The BIE (50) can be transformed into matrix form using the discretization in the BEM formulation as The stress tensor rates can be expressed as Thus, we can write ( 78) and (79) as follows Then, the boundary unknowns are where By using ( 81) and ( 82), the plastic stress rate can be expressed as where The iteration process may be summarized as follows: Step 1 Calculate d from (84); Step 2 Calculate d = ( / ) ; Step 3 Calculate d ‾ from (73); Step 4 Calculate d ‾ = d ‾   ; Step 5 Verify iteration convergence by comparing d to the value of (84) in step 1; Step 6 If the relative change between two subsequent iteration stages is less than the set tolerance (which is a value obtained by measuring a component's capacity to perform its design function in the presence of a flaw or damage), return to step 1 and start a new load increment.

Numerical Results and Discussion
The proposed BEM strategy employed in this study can be used for a broad spectrum of thermal stress sensitivity of nonlocal thermo-elasto-plastic damage problems.
To solve the systems resulting from the BEM discretization in the current study, we employed Zan et al.'s stable communication avoidance S-step-generalized minimal residual approach (SCAS-GMRES) to reduce the number of iterations and computation time [39].Table 1 compares the SCAS-GMRES [39], fast modified diagonal, and toeplitz splitting (FMDTS) of Xin and Chong [40] and the unconditionally convergent-scaled circulant and skew-circulant splitting (UC-RSCSCS) of Zi et al. [41] during our solution of the current problem.This table displays the number of iterations (Iter.),processor time (CPU time), relative residual (Rr), and error (Err.)calculated for various length scale parameter values ( = 0.01, 0.1, and 1.0).According to Table 1, the SCAS-GMRES iterative approach uses the least amount of Iter.and CPU time, meaning that it outperforms the FMDTS and UC-RSCSCS iterative methods.There were no published results that supported the suggested technique's conclusions.Some of the literature can be considered as part of the planned investigation [42][43][44][45][46].As a result, we examined a specific instance in our research and compared our BEM results to the finite difference method (FDM) and finite element method (FEM).
Figures 5-7 depict the distributions of the thermal stresses  ,  , and  overtime for the current BEM, finite difference method (FDM) of Ricci and Brünig [47], and the finite element method (FEM) of Su et al. [48].These statistics demonstrate that the BEM is in excellent agreement with the SRBNS and FEM, proving the validity and accuracy of our suggested approach.The computing results for the considered problem were obtained using Matlab R2022a on a MacBook Pro with a 2.9 GHz Core i9 processor.The boundary element technique described in this paper applies to a wide variety of nonlocal thermoelasto-plastic damage scenarios   Table 2 compares the computer resources required to model the stress sensitivity of nonlocal thermo-elasto-plastic damage problems using BEM, FDM [47], and FEM [48].This table demonstrates that the proposed BEM is more precise and efficient than both the FDM and the FEM.

Conclusions
The numerical solution of the stress sensitivity in nonlocal thermo-elasto-plastic damage problems is presented using a new boundary element method (BEM) model.A boundary element model is used to represent the numerical solution to the heat conduction equation under non-local conditions.The non-local condition specifies the overall quantity of thermal energy contained in the substance under examination.The approach for solving the heat equation will disclose an unknown control function that governs the temperature in a specific section of the solid's boundary.The initial-stress BEM for structures with strain-softening damage is used in a boundary element program with iterations at each load increment to create a plasticity model with yield limit deterioration.To eliminate the challenges associated with the numerical calculation of singular integrals, the regularization technique is applicable to integral operators.A numerical scenario is used to validate the physical accuracy and efficiency of the proposed formulation.

Conflicts of Interest:
The authors declare no conflicts of interest.

Figure 2 .
Figure 2. Variation in the thermal stress  sensitivity along the  -axis for two different softening parameters of local and nonlocal theories.

Figure 3
Figure 3 shows that the values of the stress  sensitivity in both softening parameters  = −1000 MPa and  = −3000 MPa decrease in the ranges 0 ≤  ≤ 0.2 and 0.7 ≤  ≤ 1.5 for local theory, while increasing in the range 0.2 ≤  ≤ 0.7.However, for nonlocal theory, the values of the stress  sensitivity based on softening parameter  = −1000 MPa increase in the range 0 ≤  ≤ 1 and decrease in the range 1 ≤  ≤ 1.5 , while, based on the softening parameter  = −1000 MPa, decrease in the range 0 ≤  ≤ 0.3 and increase in the range 0.3 ≤  ≤ 1.5.The values of the stress  sensitivity based on the softening parameters  = −1000 MPa and  = −3000 MPa converge to zero with increasing  for  1.5.

Figure 3 .
Figure 3. Variation in the thermal stress  sensitivity along the  -axis for two different softening parameters of local and nonlocal theories.

Figure 4
Figure 4 depicts the behavior of the values of the stress  sensitivity based on the softening parameters  = −1000 MPa and  = −3000 MPa for local and nonlocal theories, which are similar.The values of the stress  sensitivity based on the softening parameter  = −1000 MPa increase in the range 0 ≤  ≤ 1.5.However, the values of the stress  sensitivity based on the softening parameter  = −3000 MPa increase in the range 0.15 ≤  ≤ 1.5 .The values of softening parameters  = −1000 MPa and  = −3000 MPa converge to zero with increasing distance  at  1.5.

Figure 4 .
Figure 4. Variation in the thermal stress  sensitivity along the  -axis for two different softening parameters of local and nonlocal theories.

Figure 5 .
Figure 5. Variation in the thermal stress  with time for different methods.

Figure 6 .
Figure 6.Variation in the thermal stress  with time for different methods.

Figure 7 .
Figure 7. Variation in the thermal stress  with time for different methods.

Funding:
The authors did not receive support from any organization for the submitted work.Data Availability Statement: All data generated or analyzed during this study are included in this published article.

Table 1 .
Numerical results for the considered iteration methods.

Table 2 .
A comparison of the computer resources required to model stress sensitivity of nonlocal thermo-elasto-plastic damage problems.