PSI - Issue 23

Jiří Vala et al. / Procedia Structural Integrity 23 (2019) 328 – 333 Ji ří Vala, Vladislav Kozák / Structural Integrity Procedia 00 (2019) 000 – 000

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homogeneous Neumann ones). Let  consist of 2 disjoint parts (open sets again): and f  by metal fibres, with contact surfaces  : the symbol [ ]  refers to the difference between function traces on  from particular sides. It is also useful to introduce the fractured zone  of  separately. Then, using the Einstein summation indices , , , {1,2,3} i j k l  , , ( ) i  in the sense / i x   and the small strain tensors , , ( ) ( + )/2 ij i j j i v v v   , etc., the energy conservation with the unknown displacements 1 2 3 ( , ) ( ( , ), ( , ), ( , )) u x t u x t u x t u x t  , the virtual ones 1 2 3 ( , ) ( ( , ), ( , ), ( , )) v x t v x t v x t v x t  , the prescribed surface loads 1 2 3 ( , ) ( ( , ), ( , ), ( , )) q x t q x t q x t q x t  and the prescribed volume loads 1 2 3 ( , ) ( ( , ), ( , ), ( , )) g x t g x t g x t g x t  reads as (1) utilizing   * ( , )( ), 1 ( , ( , )) ( ( , )) ( ) ( ( ))d ([ ( , )])/ [ ( , )] ( )d ( ) ij ijkl kl i i A w u t v x w x t w x t C x v x x u x t u x t v x s x             , ( )( ), ( ) ( ( , )) ( ) ( ( ))d ij ijkl kl G u t v x u x t C x v x x       , ( ), ( , ) ( )d ( , ) ( )d ( ) i i i i F t v g x t v x x q x t v x s x       , (4) whereas ( , ) (0,0,0) u t   on  , compatible with a priori known initial displacements ( ,0) u  on  ; w can be taken as u here ( w u  is admitted because of the computational linearization of (2)). Serious complications are brought into (1) through constitutive relations, namely those accounting for the signifi cant irreversible processes. Seemingly (1), supplied by standard linearized strain-stress relations, working with the symmetric 4th-order tensors C of stiffness characteristics (compound from 2 Lam é constants for each material, i.e. fibres and matrix, in the simplest isotropic homogeneous case) and the scalar viscous characteristic  (often artifi cial, referring to the Kelvin viscoelastic model) can be handled using the standard theory of linear differential equa tions of evolution of parabolic type assuming zero-valued  and constant  . Consequently, using the standard nota tion of Lebesgue, Sobolev, Bochner, etc. spaces by Roubíček (2013) , taking V as the subspace of all 1,2 3 ( , ) W R    satisfying (0,0,0)   for traces on  , under reasonable requirements to such characteristics (such as positive  , positive definite C , sufficiently smooth  , some Lebesque integrability of g and Hausdorff inte grability of q in relevant function spaces, needed in embedding and trace theorems, we are able to verify the exist ence of a unique solution 2 , ( , ) u u L I V  completely, as well as the convergence of the algorithm induced by (2). Much weaker results (for more general Banach spaces) are available (and complicated, frequently non-constructive proofs occur) in more general cases; this stimulates the proper analysis of the characteristics  and  and their as sessment in (4). Here  refers to certain nonlocal damage function, zero-valued outside  , motivated by some scale bridging considerations, whereas  refers to the interface potential on  , accounting for the physical deterioration on cohesive interfaces, describing the relationship between the surface traction and material separation between the surfaces, crucial for the reliable study of matrix/fibre debonding. Since (2) refers in every time step to a linear problem, but still in an infinite-dimensional space, an additional computational discretization (except very special configurations with known analytical or semi-analytical solutions) is needed. Such discretization of (2) can be sketched as ( ) ( ) ( ) ( ) si sia a sib b sic c u x u x u x u x       , (5) containing the standard finite element shape functions ( ) a x  on  , the fractured zone enrichment shape functions ( ) b x  on  and the cohesive interface enrichment shape functions ( ) c x  on  ; a , b and c are considered as the Ein stein summation indices over all corresponding shape function sets here. The favourite choice for the implementa tion of (5) is ( ) sia si a u u x  , ( ) sib si b u u x  , ( ) sic si c u u x  (i.e. just the displacement values in selected points in the role of unknown parameters), corresponding to ( ) a x  , ( ) b x  and ( ) c x  with small compact supports near a x , b x and c x . The optimized choice of both a x , b x and c x and ( ) a x  , ( ) b x  and ( ) c x  , together with the strategy of numerical integration and mesh refinement, determine the efficiency and the robustness of relevant algorithms and stimulate the development of special problem-oriented XFEM (or similar) algorithms. m  occupied by a silicate matrix,

3. Illustrative example

For computational modelling, a specimen with cementitious matrix and steel fibres has been chosen. Numerical results demonstrate the planar crack propagation in a cracked body, depending on the fibre location and material characteristics. The stiffening effect of fibres plays a significant role for the direction of crack propagation.