  In this paper it is shown that if $\Omega \subset \mathbb{R}^N$ is an open,
bounded Lipschitz set, and if $f: \Omega \times \mathbb{R}^{d \times N \times
N} \rightarrow [0, \infty)$ is a continuous function with $f(x, \cdot)$ of
linear growth for all $x \in \Omega$, then the relaxed functional in the space
of functions of Bounded Hessian of the energy \[ F[u] = \int_{\Omega} f(x,
\nabla^2u(x)) dx \] for bounded sequences in $W^{2,1}$ is given by \[ {\cal
F}[u] = \int_\Omega {\cal Q}_2f(x, \nabla^2u) dx + \int_\Omega ({\cal
Q}_2f)^{\infty}\bigg(x, \frac{d D_s(\nabla u)}{d |D_s(\nabla u)|} \bigg) d
|D_s(\nabla u) |. \] This result is obtained using blow-up techniques and
establishes a second order version of the $BV$ relaxation theorems of Ambrosio
and Dal Maso and Fonseca and M\"uller. The use of the blow-up method is
intended to facilitate future study of integrands which include lower order
terms and applications in the field of second order structured deformations.
