We present the setup of a compact, q-switched, cryogenically cooled Yb:YAG laser, which is capable of producing over 1 J output energy in a 10 ns pulse at 10 Hz. The system's design is based on the recently published unstable cavity layout with gain shaping of the spatial intra-cavity intensity distribution. Using a hexagonal homogenized pump beam, the laser generated an according hexagonal output beam profile. The suitability of such laser properties for the intended use in a laser shock peening process is demonstrated. In the experiment an aluminum plate was treated and the generated residual stresses in the sample subsequently measured. Other applications of this laser system like laser pumping or surface cleaning are conceivable.