You can train your body to bend the same way you train it to be strong: systematically and progressively.

When I began practicing yoga thirteen years ago, the sensation of stretching was extremely uncomfortable. There are two things that have helped me significantly over the years:

Focus on which joints are involved in the action, and isolate those joints.
For instance, when I do a pancake stretch, I do not feel my hamstrings or my calves. I feel my abs. A lot. This is because I focus on flexing at the hip while keeping my legs strong. A pancake stretch is simply seated hip flexion with the legs wide. Once I am settled into the position, I breathe, inhaling for a count of four and exhaling for a count of six, for 30-60 seconds. Breathing calms the nervous system and overrides the part of your brain that says, “no! I don’t want to be here!”

Get into a position near the point of stretch and do something there.
If you are working on hip extension (pure hip extension, not lumbar extension), get into a tall half-kneeling position with your right leg back. See if you can set yourself up so you feel strong with a sense of the beginning of a stretch in your right hip. As you exhale, raise your arm overhead. Inhale, lower the arm down. Perform 6-8 repetitions and switch sides.