PSI - Issue 23

Michal Jedlička et al. / Procedia Structural Integrity 23 (2019) 445– 450 M ichal Jedlička, Václav Rek, Ji ří Vala / Structural Integrity Procedia 00 (2019) 000 – 000

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The physical analysis of cracking relies on conservation principles of classical thermodynamics, involving some micromechanical and scale-bridging considerations Ottosen & Ristinmaa (2013). In this short paper, to demonstrate some numerical and computational effects, we shall consider only the equilibrium condition (as the consequence of energy conservation)  Ω δε : σ dΩ =  Ω δ u . b d Ω +  Γ δ u . q dΓ . (2) Here Ω denotes a deformable body in the (in general) 3-dimensional real Euclidean space, supplied by the Cartesian coordinate system (where the Hamilton gradient operator  can be introduces) ; σ are the symmetric stress tensors, b the vector of volume loads, q the vector of surface loads on certain part Γ of the boundary  Ω of Ω (to incorporate the Neumann boundary conditions, if needed), u the vector of (a priori unknown) displacements (cf. (1)), δε the symmetric virtual strain tensor and δ u the vector of virtual displacements; for much more detailed explanations cf. Teng et al. (2018). T he rest of boundary Θ (  Ω = Γ  Θ, Γ and Θ are disjoint) is assumed to be supplied by the Dirichlet homogeneous boundary conditions, i. e. there are prescribed all zero values of u . For simplicity (although the following considerations can be modified to handle more general problems), because of the limited extent of this paper, we shall assume the (both physically and geometrically) linear elastic behavior, i. e. σ = D ε , 2 ε =  u + (  u ) T (3) where D contains all material characteristics, in the special case of a homogeneous and isotropic medium only 2 independent constants: the Young modulus E and the Poisson ratio  , and ε denotes t he symmetric strain tensor. Inserting (3) into (2), we are able to evaluate u theoretically; the principle difficulty is caused by the possible crea tion and propagation of cracks, driven by the stress concentration, which must be handled using (1) approximately. Particular additive terms of (2) can be then (after rather extensive calculations) expressed, element-by-element, as  Ω δε : σ dΩ  K i d i , K i = B i T D B i det J w i ,  Ω δ u . b d Ω  N i T b i det J w i , … (4) where B i contain appropriate derivatives of the nodal functions from (1), J is certain auxiliary transformation ma trix, b i refer to the values of b in selected points, whereas w i are the weights associated to the Gaussian points. The last term of (2) can be included similarly, with slight technical difficulties only (some further notations are needed). Finally (4) creates the (seemingly simple) system of linear equations K d = F (5) where K can be seen as a global stiffness matrix, F as a global nodal force vector, whereas d contains all unknown parameters (not only the standard nodal displacements, as usual in FEM), as discussed by (1). The automation of the assembly of (5) and its robust, inexpensive and effective computational analysis determines the quality of the result ing algorithm. For the following considerations let  r be the real surface crack area in some finite element and  v the hypothet ical surface across the whole element; their ration  with values between 0 and 1 will be important. The second indices in  r and  v will refer to particular finite elements. The choice of an optimal procedure for numerical inte gration handling crack initiation and propagation problems Sumi (2014) and Khoei (2015) depends both on integrat ed functions and type of elements. The Gauss quadrature is suitable for classical elements, but it is not appropriate for elements affected by discontinuities, namely for enriched elements. This difficulty can be overcome by two different approaches. The first approach is to increase the number of Gauss integration points – see Fig. 3, part a). The second approach is the method of internal meshing of elements – see Fig. 3, part b), where the element bisected by the interface is divided into triangular/quadrilateral sub-elements. An introduction of enriched trial functions into to the primary field of variables leads to the increasing number of variables in the node equations, thus such functions must be properly integrated during the process of assembly. The given function is of discontinuous character consequently is not possible to use the standard number of integra tion nodes of the Gaussian quadrature, as usual in the finite elements analysis with standard polynomial base func tions. The one of the ways to overcome this problem is a naive type of an integration algorithm which considers a 3. Computational approaches