Health　monitoring　of　structures　can　be　improved　if　a　more　universal　two-way　electromechanical　coupling,　the　flexoelectric　effect,　is　utilized.　It　is　a　coupling　between　the　electric　polarization　and　the　strain　gradients.　In　the　direct　flexoelectricity,　the　electric　polarization　is　induced　by　strain　gradients,　which　yield　also　a　finite　higher-order　stress　tensor　in　the　considered　phenomenological　theory.　Because　of　the　size-effect　in　higher-grade　theories　of　continua,　the　polarization　in　piezoelectric　solids　under　a　non-uniform　strains　in　nano-sized　structures　is　significantly　influenced　by　flexoelectricity.　This　effect　is　substantially　enhanced　near　the　crack　defects,　in　regions　with　large　strain　gradients.　The　mixed　finite　element　method　(FEM)　is　developed　from　the　variational　formulation　of　flexoelectric　boundary　value　problems.　The　C0　continuous　approximation　is　applied　independently　for　both　the　displacements　and　displacement　gradients.　The　kinematic　constraints　between　them　are　satisfied　by　collocation　at　some　internal　points　of　elements.　The　developed　computational　method　is　applied　to　general　2D　boundary　value　problems　with　cracks　under　a　dynamic　load.　The　influence　of　flexoelectricity　on　the　induced　electric　potential　and　the　crack　opening　displacement　is　investigated.　©　2020　Elsevier　Ltd