Modeling clay comes in several varieties. Some clays become hard when air dried, others need to be fired in a heat source such as an oven, and still others are designed to never harden at all. The method for drying your modeling clay, if possible, depends on the type of clay you've chosen.
Oil-Based Modeling Clay
Oil-based modeling clay, sometimes called plasticine, is meant to be worked and reworked without ever drying out. It's made by mixing natural clay with pigments and non-drying oils. This type of modeling clay is a great option for kids who enjoy the process of sculpting more than the desire to keep a finished piece. It's also used in classic stop-motion claymation film-making because the characters can be endlessly moved and posed without breaking or crumbling. There is no way to dry this type of modeling clay, so if you want to harden and preserve your work you need to choose a different type of clay.
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Never put plasticine in the oven to dry it out, since you will only create a gooey mess and possibly cause a fire.
Water-based air-dry clay is a natural modeling clay that hardens as it dries. It doesn't need to be fired or baked since the drying process happens just by leaving the clay piece out for 24 hours, or whatever time is recommended by the manufacturer. You can add water to the clay while working it, but if you make it too wet there's a risk of it cracking when it dries -- if that happens it's better to let some of the water evaporate before you start sculpting. Finish creating your piece while it's the consistency you like, and then leave it out to dry in an undisturbed place.
To repair breaking or cracking, keep an eye on your piece as it dries and fill any cracks with slip -- clay and water mixed to a creamy consistency -- as soon as you see them.
Oven-bake polymer clay will start to dry in the air, but it needs to be baked in an oven before it's completely cured. These clays, including Sculpey and Fimo, are favored for projects such as jewelry-making. In order to properly dry your polymer clay piece, you need to follow the manufacturer's firing instructions. For example, if you are using Original Sculpey, you bake your piece at 275 degrees Fahrenheit. The length of baking time is determined by the size of the piece, and in this case it's 15 minutes for every 1/4 inch thickness. Other Sculpey products and other brands of polymer clay have different firing times and temperatures, so you need to look up the specifics for your project. They are always printed on the packaging, or can be looked up on the manufacturer's website.
Sculpey University offers lessons in various sculpting techniques and project instructions that you can access at no cost.