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In each of these shape functions that is continuously differentiable, the iso-parametric mapping was implemented using the sampling points and weighting factors. This was accomplished through a transformation matrix and then using the Jacobean to perform the numerical integration. The numerical scheme that was employed for the integration was the Gauss-Legendre. This numerical integration for natural coordinates range from -1 to +1, for a function f(x) was obtain as a definite integral:
I = 1
1
f x dx
………….…………. (5.42)
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conductor flexural rigidity that is directly related to the helical strands slip. The periodic slip causes frictional energy dissipation and damping of the conductor oscillations. Thus, a better representation of damping was obtained when the point of contacts area was characterized for energy dissipation. Model results obtained using this approach was compared with experimental tests performed on the conductors, as reported in the next chapter.
5.9.1 Inter-Strand Contact Patches
Contact problems generally exhibit nonlinearity that requires numerical simulation and using the finite element method has become a common approach recently. In contact mechanics, the major challenge has to do with the determination of contact boundary condition. As opposed to the classical solid mechanics problem where the boundary conditions can be classified either as Dirichlet or Neumann boundary or mixed. The fundamental laws of physics of solid mechanics, e.g., momentum balance and mass conservation laws also apply to contact mechanics. But contact mechanics problems, are the most difficult type of problems in mechanics. This is because it involves more conceptual, mathematical and computational efforts. As shown in figure (5.4), the point of contact between strands which is similar to two contacting inclined cylinders. Contact surface FEM of beam to beam point contact can be found in [87].
The contact FEM analysis in this study was modelled using the node-to-node contact configuration. The analysis of this form of contact entails parameter such as the displacement and velocity fields as a function of the contact traction. To generate a mesh for the contact patches motion, the formulation should allow the possibility that displacement was driven by the force under a prescribed motion scheme. To implement this approach, the nodes in the contact patches is set to have a motion that implement the node-to-node contact at each time step. This can be achieved by the use of the well-known Newton-Rapson iteration scheme. The formulation of inter- strand contact was modelled with the node-to-node approach for 2D contact problems subject to finite deformation.
The node-to-node contact is maintained throughout the contact point between strands. Figure (5.6) shows a 2D contact point of two contacting strands. The normal traction N1 the compressive force always points inward and N2 opposes to the relative sliding direction. The direction of N2 implies that P1 is moving to the right side of P2. In the finite element approach to contact problems, the
“tangent traction” or the frictional force is often set to be opposite to N2. The normal traction N1 is often expressed as the product of a non-negative scalar Nc and the outward unit normal n, as displayed in Figure (5.6).
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Figure 5.6: The forces at the inter-strand contact 5.9.2 The Friction Contact Model
The consideration of friction at area of contact makes the analysis of energy conservation much more complicated at the contact patches. This is because the tangent traction depends on the normal traction and the sliding between strands is path dependent. In the present analysis, the Coulomb model of the dry friction with a constant friction coefficient μ between the friction force FT and the normal force FN in the contact point between beams was used;
N
T F
F
………….…………. (5.44) As explained earlier, the state of the interface of the contact pairs of strands at certain point can either be one of two categories: sticking or sliding. To obtain this condition, the analogy of a rigid- ideally plastic material was employed. This allows in distinguishing between two friction states:
the stick state, which is characterized by no relative displacement between the bodies, and the slip state, where the relative displacement in the form of sliding was present. The stick state will occur when the frictional traction FT is smaller than FN, and the pair of strands is in a perfect sticking condition and therefore experiences no relative motion. Using the Coulomb friction law, motion is completely prevented due to friction force being greater than the shearing force. When the imposed bending load is greater than the friction force, relative motion between the strands occurs this signifies the slip condition. To optimize this constraint imposed by the friction force, the Lagrange multiplier is used.
5.9.3 The Lagrange multiplier
Generally, a system with two-body contact problems will have the strain energy part, the kinetic energy part which is the energy contributed by the external tractions and body forces, the energy
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associated with contact forces. The balance of the global energy in a rate form can be expressed as [84]:
CONTACT E
P E K Total
E E E
E . . ………….…………. (5.45)
For the contacting surface, the FEM formulation of the kinematics and constitutive relations concerning contact can be completely modelled based on the geometric surface used for the description of the contacting bodies. In [87] the solution of the contact problem using the theory of elasticity concerning the two bodies involved and a solution is obtained by finding the minimum of the potential energy functional Π as defined:
1 2
min
min ……….. (5.46)
C
N n x x
g 1 2
Preserving the condition of non-penetrability requires that the penetration function remains non- negative
0
gN ………….…………. (5.47)
If the condition above is accompanied by the constraint of contact normal force, which can only be compressive:
0
FN ………….…………. (5.48)
0
N Ng F
As stated early, the contacting surface obeys the law of physics. The total energy in the two-body contact problem contains two parts wherein the first part comprises the kinetic energy and strain energy, and the second part comes from contribution of the contact tractions. The total energy with the imposed contact constrain is defined as:
C
Ngda
Total 1 2
………….…………. (5.49) The frequently used method to include the equality constraints is known as the Lagrange multiplier method. In this approach, the saddle-point problem of the modified functional was encountered and this leads to the stationary point formulation. This can be expressed for the contact problem as:
act N Ng
stat ………….…………. (5.50)
130 Where λ is the Lagrange multiplier
Theoretically, gN coincides with the normal contact traction N1 in the Lagrange multiplier approach [88]. If the constrain condition is satisfied exactly, the last term on the right of equation (5.48) adds nothing to the total energy. Computing the directional derivative of equation (5.49) with respect to displacement yields the stationary condition.
T
Aub
uTKuuT f T
Aub
2 1
2
1
………….…………. (5.51) The imposed constraint for the Lagrange multipliers, which constitute the set of extra unknowns obtained in the problem for the node-node contact is given as [87]:
0 0 0
0
N
NB B
NA A B
A
B N A
N
NB NBB
B NBA
NA NAB
A
R R R
R R q
q
K K
K K
K K
K K
K
………….…………. (5.52)