Our main goal is to improve, from theoretical point of view, the mechanistic understanding of bone buckling failure which is known as at the core of important clinical problems such as osteoporosis.

What is well argued is that in older bone, stability-initiated failure dominates because of the instability of the individual trabeculae which is prone to inelastic buckling at stresses far less than expected for strength-based failure. Taking advantage of our previous work, an improved original Euler's adaptive-beam buckling equation incorporating density gradient effect is investigated.

For one, we indicate that resorption can leads to new elastic instabilities that can conduct to bone-buckling mechanism of fracture. For another, we demonstrate that bone density gradient play a key role in the initiation of the bone-column elastic buckling instability.

As a result, it is clearly stated here that firstly, the number of these elastic instabilities which are potentially implied in the mechanisms of bone fracture, localized at the trabecular element scale, depends strongly upon the material parameter η and secondly; the bone density gradient affect notably the stability of the bone-column deflection.