PAMM · Proc. Appl. Math. Mech. 10, 177 – 178 (2010) / DOI 10.1002/pamm.201010081
Contact interaction inside knot connections of cables Alexander Konyukhov1,∗ and Karl Schweizerhof1,∗∗ 1
Institut für Mechanik, Karlsruhe Institute of Technology, 76128, Karlsruhe, Kaiserstraße 12
Knots as a method for the fastening of ropes and other linear materials are widely appearing in practical applications – in sailing, in surgery, in textile and rope structures etc. The mechanics of knots, however, appears to be not sufficiently covered neither by analytical methods, nor by computational methods. From a computational mechanics point of view a knot is a perfect example requiring both a robust smooth cable element and a robust curve-to-curve contact algorithm. The current contribution is aimed on the development this combination – the isogeometric approach for curvilinear beams and the robust curve-to-curve contact algorithm for curvilinear cables. The developed model is applied studying the mechanics of various knots. c 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
1
A FE model as combination of curve-to-curve contact and curvilinear beam model
Among the possible finite element models for discretization of continuum and contact modeling the following combination is optimal for cable contact: • Curve-To-Curve contact model as developed in [1] and • finite beam element model allowing finite rotations as in [2] together with isogeometric enrichment allowing C1-continuity, see [3].
Fig. 1 Curve-to-curve contact model. Definition of curvilinear coordinate systems attached to the curves.
1.1 Curve-To-Curve contact model The curve-to-curve contact element is considered equivalently in each curvilinear coordinate system corresponding to the Closest Point Projection (CPP) procedure on a curve, see Fig. 1. ρ2 (s1 , r, ϕ1 ) = ρ1 (s1 ) + re1 (s1 , ϕ1 ); e1 (s1 , ϕ1 ) = ν 1 cos ϕ1 + β 1 sin ϕ1 1 ⇆ 2.
(1)
Here, the vector ρ2 is a vector describing a contact point on the second curve, ρ1 (s1 ) is a parameterization of the first curve; a unit vector describing the shortest distance e1 (s1 , ϕ1 ) is written via the unit normal ν 1 (s1 ) and the bi-normal β 1 (s1 ) of the first curve. Eqn. (1) describes the motion of the second contact point in the coordinate system attached to the first curve. The description is symmetric with respect to the choice of the curve 1 ⇆ 2. Convective coordinates are used as measures: r – is mutual for both curve and a measure for normal interaction; sI – for tangential interaction and ϕI – for rotational interaction for the I-th curve. In the case of curve-to-curve contact there is no classical “master” and “slave” parts and both curves are equivalent. The model allows to define arbitrary cross-sections in orthogonal planes ν 1 β 1 and ν 2 β 2 . The weak form expressing equilibrium between curves is written in the symmetric form n p (2a) δW = Sym T1 (1 − rk1 cos ϕ1 )2 + (rκ1 )2 δs1 + N δr + M1 δϕ1
∗ ∗∗
κ1 r 2 +T1 p δϕ1 + M1 κ1 δs1 (1 − rk1 cos ϕ1 )2 + (rκ1 )2
)
,
(2b)
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c 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
178
Section 4: Structural Mechanics
where the operator of symmetry is used in the sense Sym{T1 } ≡ 21 (T1 + T2 ) to express the virtual work from both curve segments being in contact. The parameters in eqns. (2a-2b) are as follows: T1 , M1 are tangential force and rotational (torsional) moment between the curves applied to the first curve, N is a mutual normal force acting between the curves; k1 is the curvature and κ1 is the torsion of the first curve. Due to the symmetry operator all parameters are summed up for the second curve as well. 1.2 Curvilinear finite element beam model A special finite beam element formulation allowing both finite rotations and enrichment with arbitrary curvilinear geometry, see Ibrahimbegovic [2], is taken. The kinematics of deformation is based on the Timoshenko hypothesis. A weak form expressing the equilibrium of the beam is written Z Z 1 1 Wint = ǫT n + κT m ds0 = ǫT ΛCn ΛT ǫ + κT ΛCm ΛT κ ds0 , (3) 2 L 2 L where ǫ is a strain measure for the mid-line of the curvilinear beam, κ is a bending strain for the curvilinear beam. Finite rotations are represented by the matrix Λ computed via quaternions. The matrix Cn represents the stiffnesses of the mid-line with regard to a load vector n containing the axial and two shear forces. The matrix Cm represents the bending and torsional stiffnesses with regard to a moment vector m containing torsional and two bending moments. As an example of a finite element approximation allowing C 1 -continuous isogeometric representation of the curve a cubic spline of Hermite type can be taken. It leads to the following finite element with Hi3 (t) Hermite polynomials and x(i) position vectors: H23 (t) H13 (t) H 3 (t) H 3 (t) 3 (1) 3 H0 (t) − x(t) = x + x(2) H3 (t) + − x(3) 1 + x(4) 3 . (4) 2 2 2 2 A contact element is then considered as two opposite spline segments from the potentially contacting curves defining a beam-to-beam contact pair leading to a contact element possessing twice as many nodes as the corresponding beam element.
2
Numerical example – Square knot
The Square knot is modeled as follows, see Fig. 2: A spline forming a loop is passing through 33 characteristic nodes. Thus, each cable is modeled with 32 C1-smooth spline beam elements. Two loops are positioned initially without contact to form an opened Square knot, see Fig. 2. The material is linear elastic with E = 200, ν = 0.3. The cross section of both cables is circular with radius r = 1. Dirichlet boundary conditions are applied at points A and B in order to supply symmetry boundary conditions in plane XOZ. The displacement vector along the OX-axis is applied incrementally at both ends of both cables.
symmetry BC
30
20
u
15
A
20
10
Z
B
u 10
5 −30 −20
0
0 −20
−15
X
−5
−10 −5
Y
O
−10
−10 −10 0
0 5
10 −20
−15
10 20
15 −20 20 −25
−20
−15
−10
−5
0
5
10
15
20
25
−30
30 −30
−20
−10
0
10
20
30
Fig. 2 Geometry of a knot. Reference configuration with symmetric boundary conditions (left) and deformed configuration (right).
During modeling, see reference and deformed configuration in 2, it has been found that only C1-smooth spline finite elements are capable to represent the result and achieve convergence. Higher order finite elements (quadratic, cubic), but without C1-continuity are showing disconvergence during the first crossing of element boundaries even for reasonably small step size.
References [1] A. Konyukhov, K. Schweizerhof Geometrically exact covariant approach for contact between curves. Comput Method Appl M, accepted, doi:10.1016/j.cma.2010.04.012. [2] A. Ibrahimbegovic On finite element implementation of geometrically nonlinear Reissner’s beam theory: three-dimensional curved beam elements. Comput Method Appl M, 122:11–26, (1995). [3] J.A. Cottrell, T.J.R. Hughes, Y. Bazilevs Isogeometric Analysis Toward Integration of CAD and FEA (Wiley, 2009), p. 352. c 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
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