Since　the　advent　of　finite　element　methods,　the　dynamic　response　analysis　of　solids　and　structures　follows　such　a　route　without　exception.　Firstly　the　spatial　discretization　is　carried　out　and　the　system　of　second　order　ordinary　differential　equations　with　the　degrees　of　freedom　as　the　unknown　functions　of　time　is　derived,　which　is　called　the　semi-discrete　scheme.　Then　the　temporal　discretization　is　performed　to　the　system　of　ordinary　differential　equations　and　the　system　of　algebraic　equations,　referred　to　as　the　fully-discrete　scheme,　is　obtained.　This　route　has　been　working　well　for　most　problems,　where,　the　meshes　deform　continuously　and,　in　all　the　time　steps,　all　the　degrees　of　freedom　are　valid　and　the　number　of　them　keeps　invariant.　In　the　simulation　of　crack　propagation,　however,　even　the　number　of　degrees　of　freedom　varies　with　crack　propagation　and　those　degrees　of　freedom　associated　with　crack　tips　become　meaningless　after　the　crack　tips　move　away.　While　this　causes　no　difficulties　in　linear　static　solutions,　it　is　not　readily　handled　in　time-dependent　solutions,　leading　to　the　transfer　issue　of　degrees　of　freedom.　Opposite　to　the　conventional　order　of　discretization,　in　this　study　the　temporal　discretization　is　put　prior　to　the　spatial　discretization.　In　this　way,　all　the　degrees　of　freedom　are　valid　only　within　the　current　time　step.　The　transfer　issue　of　degrees　of　freedom　is　accordingly　resolved　elegantly.　The　implementation　of　the　proposed　procedure　is　in　the　framework　of　the　numerical　manifold　method,　illustrated　by　some　typical　examples,　where　compressed　and　sheared　cracks　are　involved　with　frictional　contact.