What is claimed is:1. A display device comprising:a flexible substrate comprising:a display area; anda peripheral area outside of the display area;an inorganic layer on the flexible substrate;an opening pattern formed in the inorganic layer and corresponding to the peripheral area;an insulating layer disposed on the inorganic layer and corresponding to the peripheral area;a first organic material layer disposed on the flexible substrate and covering the opening pattern;a first dam portion spaced apart from the opening pattern and disposed between the opening pattern and an edge of the flexible substrate;an organic light-emitting diode over the inorganic layer; anda thin film encapsulation layer dispose on the organic light-emitting diode and the thin film encapsulation layer including at least one organic encapsulation layer and at least one inorganic encapsulation layer:wherein the thin film encapsulation layer does not overlap the opening pattern.2. The display device of claim 1, wherein a portion of the inorganic layer is removed to form the opening pattern.3. The display device of claim 2, wherein the first organic material layer is directly on a portion of the flexible substrate from which the portion of the inorganic layer is removed.4. The display device of claim 1, wherein the opening pattern has a slit structure corresponding to the peripheral area.5. The display device of claim 1, wherein the opening pattern has a grating structure corresponding to the peripheral area.6. The display device of claim 5, wherein the grating structure includes a portion of the inorganic layer and the insulating layer in a center portion of the opening pattern.7. The display device of claim 1, further comprising:thin-film transistors over the flexible substrate;a first planarization layer disposed on the thin-film transistors; anda second planarization layer on the first planarization layer.8. The display device of claim 7, wherein the first organic material layer and the second planarization layer include a same material.9. The display device of claim 1, wherein the first organic material layer is disposed on the first dam portion.10. The display device of claim 9, wherein the first dam portion includes a first insulating layer on the inorganic layer, a second insulating layer on the first insulating layer, and a third insulating layer on the second insulating layer.11. The display device of claim 9, wherein the first dam portion includes a first insulating layer directly on the inorganic layer, a second insulating layer directly on the first insulating layer, and a third insulating layer directly on the second insulating layer.12. The display device of claim 1, wherein the first organic material layer covers an entirety of the first dam portion.13. The display device of claim 1, wherein the first dam portion includes a first dam and a second dam spaced apart from the first dam.