Informaldescriptionofdesignpatternswasadoptedtofacilitatetheirunderstandingbysoftware developers. However, these descriptions lead to ambiguities limiting their successful use in real applications.Recently,severalformalmethodshavebeenemployedtoguaranteeprecisespecification ofdesignpatternelementswithoutinsistingontheirbehavior.Themaingoalofthisarticleisto propose a rewriting logic-based framework for enhancing formal description and reasoning on designpatternscomposition.TheproposedmodelisintegratedandexecutedunderMaudeusing K-Maudetool.Itallowsfirsttoassociateametamodelnotation,accordingtoMDAapproach,to alldesignpatternselementsandthen,aMaudesemanticbasisforspecifyingboththeirstructural anddynamicaspects,isgiven.Inparticular,itwillbeshownhowtheauthors'combinedmodeling approachisexploitedfordefiningasetofoperationsonpatterns,andhowitwillbeusedtoformal analyzepatternscomposition.