Device-to-device　(D2D)　communication　is　one　of　the　promising　technologies　in　the　5th　generation　of　wireless　communication　(5G).　This　paper　investigates　the　achievable　transmission　capacity　for　multi-mode　D2D　communication.　By　utilizing　stochastic　geometry　tools,　we　first　provide　the　system　model　and　derive　the　transmission　capacity　for　two　different　sharing　schemes　of　underlaying　D2D　mode.　The　first　one　is　that　each　cellular　spectrum　can　be　reused　by　single　D2D　pair　and　the　other　one　considers　the　multiple　sharing　among　several　D2D　pairs　with　the　same　spectrum.　In　addition,　we　analyze　the　strategy　for　the　optimal　selection　between　both　different　sharing　scenarios.　Secondly,　we　analyze　the　transmission　capacity　for　overlay　D2D　mode.　Finally,　with　the　optimal　relay　selection　strategy,　the　maximization　of　transmission　capacity　for　relay-assisted　D2D　communication　is　obtained　under　the　constraint　of　minimum　energy　consumption.　Furthermore,　simulations　are　carried　out　and　the　results　demonstrate　the　effectiveness　and　validity　of　our　analysis.　(C)　2017　Published　by　Elsevier　B.V.