2D elastoplastic boundary problems solved by PIES without strongly singular surface integrals

doi:10.1016/j.euromechsol.2017.04.001

European Journal of Mechanics - A/Solids

Volume 65, September–October 2017, Pages 233-242

https://doi.org/10.1016/j.euromechsol.2017.04.001 Get rights and content

Highlights

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The approximation strategy for plastic strains in PIES method is proposed.
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Calculation of strongly singular surface integrals has been eliminated.
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They are replaced by simple interpolating polynomials and their derivatives.
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The strategy is effective since the domain is modeled globally by surface patches.
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Solutions obtained are very close to analytical results.

Abstract

The paper presents the parametric integral equation system (PIES) without strongly singular surface integrals in elastoplastic boundary value problems. Plastic strains in PIES are approximated by interpolating polynomials and their derivatives instead of using the integral identity. Moreover, in the proposed method a boundary and a domain are not discretized by elements and cells, but are defined globally by the smallest number of curves and surfaces. Several examples are solved. The results are compared with exact values, numerical solutions obtained by other methods and also with PIES solutions obtained by the version with the singular integral identity. The results presented confirm the reliability and accuracy of the proposed approach.

Introduction

Real phenomena in the deformable bodies are most often nonlinear. Therefore, the methods used for solving such problems are highly required. Lack of analytical approaches leads to developing of numerical. However, solving nonlinear problems, even numerically, is difficult. This is related to their nature i.e. dependency on the unknown quantities, which causes the necessity of using iterations. Another problem is the convergence, which depends on a good initial guess.

There are two most popular numerical methods for solving nonlinear boundary value problems. The first and more general one is the finite element method (FEM) (Zienkiewicz, 1977, Nguyen-Xuan et al., 2016, Ameen, 2005). Its main feature is modeling a domain and a boundary by finite elements. Such approach has advantages and drawbacks. The strong point is the possibility of modeling any shapes, while the difficulty is the effort needed to define the large number of elements. The second commonly known method is the boundary element method (BEM) (Ameen, 2005, Gao and Davies, 2002, Telles, 1983, Aliabadi, 2002). It differs from FEM mainly in the way of discretization, which in BEM is restricted only to the boundary. However, this is true only for problems without the necessity of defining the domain (e.g. elasticity without body forces). In other cases (e.g. plasticity, elasticity with body forces) the domain has to be modeled using so-called cells and this process, in practice, is similar to discretization in FEM.

Loss of the main advantage of BEM has become a major cause for commencing the research by the team to which the author belongs. The presented in this paper approach called the parametric integral equation system (PIES) is still dynamically developed in various directions. PIES has been tested on linear problems modeled by Navier (Zieniuk and Boltuc, 2006a), Laplace (Zieniuk and Szerszen, 2014) and Helmholtz (Zieniuk and Boltuc, 2006b) equations, and the author made an attempt to apply it also to elastoplastic problems (Bołtuć, 2016). The most important features of PIES, which demonstrate its efficiency are: lack of classical discretization of both a boundary and a domain, separation of approximation of a shape from boundary functions, easy modification of the defined geometry. How these benefits were obtained? PIES is the analytical modification of the classic boundary integral equation (BIE). This modification bases on including curves and surfaces known from computer graphics (Farin, 2002, Salomon, 2006) directly into kernels of PIES, in other words to mathematical formalism of PIES. Thus, the accuracy of the shape definition and the accuracy of solutions are provided independently. From that moment, a boundary and a domain can be defined more globally using the smallest number of input data. Moreover, the approximation of solutions can be also done in more efficient way. Another advantage is that modification of the defined shape can be carried out by the small number of control points, and is much easier than in FEM or BEM. Since in PIES, like in BEM, only a yield region (unknown a priori) is defined, it is very beneficial if it can be modeled and modified by the small number of data. If zones with different size and shape can be modeled using the same number of points, it is obvious to define the bigger one, which is also safer. In BEM, modeling bigger zones means more effort, because of more cells.

However, PIES like BEM has one serious drawback. To solve elastoplastic problems strongly singular surface integrals have to be calculated (Telles, 1983, Gao and Davies, 2002, Aliabadi, 2002). It requires special procedures, effort and can be very tedious. Therefore, in this paper the author proposes replacing of the singular integral identity by the approximation strategy for derivatives of displacements. The main aim of this approach is to interpolate displacements by any appropriate technique, differentiate the obtained interpolation polynomial in respect to required variables and then using stress-strain relationship to calculate stresses at any point of the domain and the boundary. Finally, plastic strains can be approximated iteratively using calculated components of a stress tensor.

The main aim of this paper is to developed and test the numerical approximation strategy for plastic strains. The Lagrange polynomial is used to interpolate displacements, while Bézier surfaces to model the plastic zone. Three examples are considered. In two of them results obtained by the proposed approach are compared with the analytical solutions, while in the last one with numerical solutions derived by FEM and BEM. The results from PIES with the singular integral identity are also included in all cases.

Section snippets

The Navier rate equation

The equilibrium equation in incremental form is given as follows (Aliabadi, 2002) ${\dot{σ}}_{i j, i} + {\dot{b}}_{j} = 0,$ where ${\dot{b}}_{j}$ are body force rates and ${\dot{σ}}_{i j}$ stands for components of the stress rates. Time derivatives are indicated by a dot above variables, while space derivatives by a comma.

Assuming infinitesimal strain theory, the total strain-displacement relationship in terms of rate quantities is given by ${\dot{ε}}_{i j} = \frac{1}{2} ({\dot{u}}_{i, j} + {\dot{u}}_{j, i}) .$

The total strain rate ${\dot{ε}}_{i j}$ is expressed as a sum of elastic and plastic components as

Interior values of the stress and strain rates

As was mentioned above, the stress integral identity contains strongly singular surface integral, which is difficult to calculate accurately and effectively. Therefore, the author has attempted to replace such approach by a simpler algorithm with the displacement integral identity only. The idea bases on calculating displacement rates at interpolation nodes using the non-singular integral identity. Then these values are used to obtain the interpolation polynomial, which should be differentiated

Algorithm of nonlinear solution

In order to solve PIES (7) values of ${\dot{ε}}^{p}$ and $Δ {\dot{ε}}^{p}$ (increment of plastic strain) are required. The first step is to assume zero plastic strain rates everywhere. It results in elastic solution and possibility of obtaining the load at first yield (by comparing the equivalent stress of the highly stressed node $σ_{e}^{\max}$ with the uniaxial yield stress of the material $σ_{Y}$ ). As is known, the solution is found in the incremental process, which starts from reducing stress value with a load factor ( $λ_{0} = \frac{σ_{Y}}{σ_{e}^{\max}}$

Example 1

A first considered example concerns a simple square geometry shown with boundary conditions in Fig. 5. The response of the unit square subjected to uniaxial tensile displacement has been analyzed. The problem is considered as plane stress, while as a yield criterion the Von Mises criterion with linear hardening is assumed. Moreover, the following properties have been established: $E = 1 M P a$ , $ν = 0.3$ , $H^{'} = 0.1$ and $σ_{Y} = 0.8 M P a$ . Calculations have been performed in six increments with tolerance 0.00001.

The

Conclusions

The paper presents the numerical strategy for approximating plastic strains. The proposed approach was developed in order to replace the use of strongly singular integral identity, which is required for calculating stresses within the domain. Approximation is performed by Lagrange polynomials, which are easily differentiable. The plastic zone is modeled by Bézier surfaces. It gives opportunity for simple arranging of interpolation nodes, and also for easy modification of plastic regions.

Three

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PIES for 2D elastoplastic problems with singular plastic strain fields
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On the other hand, they are not required in that quantity for modeling the shape, because it is often elementary, but has, for example, a notch. An alternative method, the parametric integral equation system (PIES) [9–12], is known from the fact that it separates the approximation of the shape from the approximation of the solutions. It is possible because the shape is analytically integrated into the mathematical formula of PIES.
The paper presents the approach for solving 2D elastoplastic problems with singular stress/strain fields by the parametric integral equation system (PIES) method. PIES is applied since it does not require the discretization of the plastic zone into elements, and instead, it uses globally declared parametric surface patches. The properties of such surfaces (e.g. the unit square is their domain) are used for automatically concentrating the interpolation nodes around the regions with singular stress/strain fields. This, in turn, limits the global number of nodes in favor of their accumulation in specific places. The proposed approach of node concentration requires only a few parameters to be defined. In order to take into account the complex nature of the plastic strain fields, a special method of their interpolation is selected. The Kriging approach effectively involves an interactive investigation of the spatial behavior of the strains to choose the best estimate. It is also the method that bases on irregularly spaced interpolation nodes. Some examples are solved, and obtained results are compared with the finite element method (FEM).
A separation of the boundary geometry from the boundary functions in PIES for 3D problems modeled by the Navier–Lamé equation
2018, Computers and Mathematics with Applications
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The results obtained in this way are highly accurate, especially in the vicinity of the boundary. In the case of plastic problems, this strategy also eliminates the need to calculate strongly singular integrals over the domain [26]. The performed tests show that the number of input data in PIES is significantly lower compared to FEM and BEM with the possibility of defining the boundary in a continuous way.
In this paper, we present a modification of the Somigliana identity for the 3D Navier–Lamé equation in order to analytically include in its mathematical formalism the boundary represented by Coons and Bézier parametric surface patches. As a result, the equations called the parametric integral equation system (PIES) with integrated boundary shape are obtained. The PIES formulation is independent from the boundary shape representation and it is always, for any shape, defined in the parametric domain and not on the physical boundary as in the traditional boundary integral equations (BIE). This feature is also helpful during numerical solving of PIES, as from a formal point of view, a separation between the approximation of the boundary and the boundary functions is obtained. In this paper, the generalized Chebyshev series are used to approximate the boundary functions. Numerical examples demonstrate the effectiveness of the presented strategy for boundary representation and indicate the high accuracy of the obtained results.
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View full text

2D elastoplastic boundary problems solved by PIES without strongly singular surface integrals

Highlights

Abstract

Introduction

Section snippets

The Navier rate equation

Interior values of the stress and strain rates

Algorithm of nonlinear solution

Example 1

Conclusions

Eng. Anal. Bound Elem.

Appl. Math. Model

Appl. Math. Comput.

Eur. J. Mech. A-Solid

Math. Comput. Model

Int. J. Solids Struct.

The boundary element method

Computational Elasticity

Mathematical Analysis

The Boundary Element Method with Programming. For Engineers and Scientists