I would like, in order to prevent frays, to glue the sides and the thread stops of a finished work. The work should then be washable.
Which kind of glue should l use?
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You could try fray check. These glues tend to be thinner, which would give you a less noticeable and stiff edge. They are usually clear, but may yellow over time.
If that isn't strong enough, you could try fabric glue, which as the name suggests, is specifically made for bonding fabrics.
You can also finish the back of your piece without glue! After mounting on a board or in a hoop, fold the raw edges under and stitch across from one side to another, keeping the fabric taut and in place. You can then cover up the back by sewing on another piece of fabric. This is often done with felt which does not fray, but you could use any fabric as long as you tuck the raw edges under.
For my cross stitch work, I machine stitch the edges all around with a zigzag (preferably a three-step). If by hand, I use a scaled down version of a blanket stitch. For something that will be washed, I would make the stitches closer though not as close as a buttonhole stitch.
As for worries about stitches coming loose, I have none. I learned as a youngster how to begin and end a stitching line (seam or embroidery) securely and invisibly (for nearly all items). I leave the at least a needle's length of thread under an already stitched area, then stitch over where the needle came up about three times but not in the exact same place--angle your hand differently and use a different thread on the fabric each time. (Wish I had a picture; this sounds harder than it is.) I end the same way. I've used this technique on everything from wedding dresses to tents and never lost a bride or a camper.
This craftsy link may help with other ways to fasten your thread. https://www.craftsy.com/blog/2015/05/how-to-start-embroidery-without-a-knot/ These techniques are also fantastic.
Personally, I would stay away from any form of glue or tape. I've tried it in years past and found the results to be far less than satisfactory and unpleasant to work with during the stitching. And glue does deteriorate over time much faster than thread. And the sticky goop it leaves behind makes for a long cleaning story. ugh...
You could also try using fusible hemming tape or fusible interfacing. You could then prepare the edges either by laying the hemming tape about 1/2" from the edges of your cross stitch fabric, and then folding the edge down and adhering the whole thing with your iron (follow the instructions for hemming tape, and you will see what I mean. Or you could use a wide strip of fusible interfacing and fold it over the edge of the cross stitch fabric, so that the fold in the interfacing is protecting the edge. Both of these types of material are designed to remain fixed in place with washing, and I don't trust fabric glue quite as much. It does mean that your cross stitch fabric has to be able to withstand an iron--if it is standard Aida cloth it should be fine, but if you are using fabric with some sort of coating, I would test a little bit before ironing.
By the way...I have never tried this, but there is no reason you couldn't fix the edges in this way before beginning to work your cross stitch pattern. As long as you are using interfacing or tape that is the correct weight for your fabric (slightly lighter weight than the fabric itself), it shouldn't interfere with using an embroidery hoop, and will keep the edges neater throughout the project.