Computer Methods in Applied Mechanics and Engineering
Fast semi-analytic computation of elastic edge singularities
Introduction
In linear elasticity, problems are usually solved with industrial codes using the finite element method. However, when the elastic body has corners on its boundary, such as a polygon or a polyhedron, the solution obtained is inaccurate near the corners. The reason is that on such domains, elliptic boundary value problems admit singular solutions. Thus, special “tools” to acquire some knowledge about those solutions can be very useful.
We consider the linear equilibrium equations of an elastic material , possibly heterogeneous: we suppose that can be decomposed into several homogeneous parts , with k=1,…,K. Here we treat corners in two-dimensional domains and edges in three-dimensional ones. The corners and edges in question are those of , of course, but also those of any of the homogeneous subdomains . The equations in can be written in the general formwith boundary conditions on and transmission conditions at the interfaces between the .
If is two-dimensional, from the general theory (see [4], [5], or [7]), we know that, near any of its corners , the solution can be seen as a sum of a regular part ureg and a singular one using. Except in particular cases where logarithmic terms appear, using can be expanded in the generalized formwhere (r,θ) are the polar coordinates with as origin, Bi the stress intensity factors, νi the singularity exponents (or eigenvalues) and gi(θ) are the angular singular functions (or eigenfunctions).
The numbers νi are real or complex and characterize, along with the associated angular functions gi, the behaviour of the solution near . They only depend on the boundary conditions, on the geometry of the subdomains , and on the material laws.
In the case of an edge, the description of the splitting into regular and singular parts is more complicated, see [3] for instance, but still involves singularity exponents νi and angular singular functions gi, possibly changing along the edge.
We are interested here in the computation of those two quantities, taking into account expressions provided by the theoretical work [1]. The core of the method is the knowledge for each and each material index k, of an explicit solution basis of the homogeneous system without boundary conditionsApplying the boundary and transmission conditions to these bases leads to the construction of a matrix A(ν) whose determinant is called “characteristic determinant”. Singularity exponents are the roots of the equation det A(ν)=0, whose solution is obtained using Cauchy integrals.
Such a method is known and has been put into practice for isotropic materials, see [8]. But up to now, the other methods proposed for anisotropic materials are more numerical and less analytic than ours.
By a finite element approach in the one-dimensional domain Θ of the angular variable θ, Leguillon and Sanchez-Palencia in [6] construct a matrix whose eigenvalues are the singularity exponents. Yosibash [11], [12] uses a formulation of the problem based on a modified Steklov method and constructs also a matrix with a similar role, by the p-version of finite elements in a thin bi-dimensional annulus of the form .
The method of Papadakis and Babuška [10] is closer to ours: they solve numerically Eq. (3) with transmission conditions and boundary conditions on one side by two initial value problems and construct a matrix with the boundary conditions on the other side and then find the roots of the characteristic determinant det using Cauchy integrals.
The main advantage of our method is to remain as close as possible to the exact solution since the solution bases of the equation are known almost analytically (only the roots of a symbol associated with the system Lk have to be computed). In particular, the angular singular functions gi belong to the spaces generated by the above mentioned bases. We stress again the fact that an important part of the calculation is made explicitly, which is essential considering accuracy and computation time.
In 2 Theoretical aspects: the solution bases, 3 Expression of the elasticity system, we recall from [1], [2] the determination of the solution bases of Eq. (3). In Section 4 the principles for the construction of the matrix A(ν) are explained and in Section 5 their numerical implementation is described. In 6 Computation of singularity exponents, 7 Computation of singular functions, we present the computation of the singularity exponents and of the angular singular functions. In Section 8 we treat some specific aspects of three-dimensional geometries. Finally, to show the range of application of the method and compare with the earlier works, we give in Section 9 various examples in two and three dimensions, especially for anisotropic materials, and with different boundary conditions. We draw some conclusions in Section 10.
Section snippets
Theoretical aspects: the solution bases
In this section and in the following one, we concentrate upon the solution basis of Eq. (3) for one material. Thus, we drop the material index k.
Expression of the elasticity system
Notation. In the following, we write ∂k instead of ∂xk.
The system L comes from the application of the laws of mechanics in the case of linear elasticity and depends on the classical quantities:
- •
displacement field u=(u1,…,ud),
- •
strain tensor ,
- •
stress tensor σij(u)=∑klaijklεkl(u), where aijkl are elasticity constants satisfying the following symmetry properties:
The general equilibrium equation isor, taking into account the
Characteristic determinant
We already know a solution basis of the homogeneous problem without boundary or transmission conditions. To compute the singularity exponents, we have now to take them into account.
Computation of the characteristic matrix
We describe here the computational aspects of the determination of the matrix A(ν).
Numerical aspect
Singularity exponents are the roots of the equation det A(ν)=0. To compute this determinant, we make an LU decomposition of A(ν).
The next problem is to locate the roots. For that purpose, we use Cauchy integrals over closed curves. Given holomorphic functions f and g and a closed simple contour C, Cauchy's formula iswhere S is the set of the roots of f in the interior of C. Taking g(ν)=1, we can compute the number of roots inside the curve and thus isolate each
Computation of singular functions
Let be defined as ϕ(k)j(ν,θ) for θ∈(ωk−1,ωk) and by 0 for θ∈(ω0,ωK)⧹(ωk−1,ωk). A singularity exponent ν is solution of the equation det A(ν)=0, that is to say there exists a non-zero linear combination, cf. (7),solution of the homogeneous problem with boundary conditions (6): any such function is an angular singular function associated with the singularity exponent ν.
Coefficients z(k)j are the components of a vector z lying in ker A(ν). In order to
Case of a three-dimensional domain
We know that in a point lying on an edge, singularity exponents do not depend on tangential derivatives ∂z. Thus, the problem remains bi-dimensional in the plane . Computing singularity exponents in other points of the edge means to move the local axes along the edge and the coefficients of rigidity matrices in the local basis, because, except for isotropic material, those coefficients depend on the basis used.
Each material is assumed to be homogeneous, so coefficients are invariant
Examples
In this part, we illustrate the efficiency of the method. We compare with known results, focusing on some interesting details. At last, we propose a typical example of our own, showing what is easy to obtain in dimension 3.
All the computations have been done on a DEC Alpha workstation using double precision arithmetic. The tolerance ε had been set to 10−9.
Conclusion
In this paper, we have presented a method for the computation of singularity exponents in linear elasticity, which is especially useful in the anisotropic case. The method is based on the construction of a matrix of low dimension depending on a complex variable ν, whose determinant is 0 for a discrete set of values of this variable. These values are the exponents of singularities.
Except during the preliminary step which consists of the computation of the roots of a certain polynomial associated
References (12)
- et al.
A finite element approach to three-dimensional singular stress states in anisotropic multi-material wedges and junctions
Int. J. Solids and Structures
(1996) - et al.
A numerical procedure for the determination of certain quantities related to the stress intensity factors in two-dimensional elasticity
Comput. Methods Appl. Mech. Engrg.
(1995) - et al.
Construction of corner singularities for Agmon–Douglis–Nirenberg elliptic systems
Math. Nachr.
(1993) - et al.
Computation of corner singularities in linear elasticity
Lect. Notes in Pure and Appl. Math.
(1994) - M. Dauge, Elliptic Boundary Value Problems in Corner Domains – Smoothness and Asymptotics of Solutions, Lecture Notes...
- P. Grisvard, Singularities in Boundary Value Problems, RMA, vol. 22, Masson, Paris, France,...
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