Explicit fourth-order stiffness representation in non-linear dynamics

In momentum-based time integration methods, the internal forces appear naturally as an approximate representation of the time integral of the internal forces over the integration interval. It is highly desirable that this force integral also represents the increment of the internal energy. A simple global form of the effective internal force is presented, in which it is represented by its algebraic mean value plus a higher order term in the form of the product of the increment of the tangent stiffness matrix at the interval end-points and the corresponding displacement increment. This explicit representation is of fourth order, and leads to the exact energy increment for systems with quartic internal energy function.