ARUP; ISBN: 978-0-9562121-5-3 - CMBBE 2012 - Cardiff University

h a =acetabular cartilage
d = joint space
h f =femoral cartilage
spring length = h a + h f + d
spring compression = ∆x
spring force = k∆x when ∆x > d
Figure 1. Sagittal view of DEA representation. Bones were rigid and cartilage was represented
by an array of springs (left). 3D FE model; triangular shell and hexahedral elements defined
cortical bone and cartilage, respectively (right).
the corresponding location. Since the spring attachment at the femoral side did not
necessarily coincide with a surface node, the femoral cartilage thicknesses were
interpolated from neighboring nodes on the femoral surface. Joint space for each spring
was calculated by subtracting the sum of the acetabular and femoral cartilage
thicknesses from the initial spring length. The springs resisted compressive forces
(spring length less than the sum of acetabular and femoral cartilage thickness) but not
tensile forces (Fig. 1). The force generated by compression of an individual spring, f i
was calculated according to Hooke’s law:
fi kixini , [1]
where ∆xi = distance of spring compression, ki = spring stiffness, and ni = local surface
normal. The spring stiffness, ki, depended on Young’s modulus and Poisson’s ratio (E =
11.85 MPa, v = 0.45) [7, 8] and was defined as:
E(1 v) Ai
ki
. [2]
(1 2 v)(1 v)
h
i
Here Ai is an element area and hi is the sum of acetabular and femoral cartilage
thicknesses. The spring forces (Eq. [1]) that balance the applied force are a nonlinear
function of the femur position. To solve this nonlinear problem, Newton’s method was
used to determine the position of the femur so that the spring forces balance the applied
force exactly. At each Newton iteration, projection points of
springs on femoral head were re-generated to account for the
updated position of the femoral head. Contact stresses were
calculated from the spring force and element area where each
spring was attached.
A convergence study was performed to determine the number
of springs necessary. The DEA modeling method was also
verified by comparisons to the analytical solution for a classic
boundary value problem [12, 13] (Fig. 2). DEA contact
stress predictions were compared to the analytical solution of
an elastic sphere supported bilaterally by concentric rigid
spheres [12, 13].
Rigid
Deform.
Rigid
24mm
20mm
Figure 2. Schematic of
verification problem
under 2000 N load.
θ