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An approach for nonlinear contact surface analysis and application to pile installation

This paper presents a simple and physical approach for analysis of contact problems between rigid and deformable bodies. The basis of the approach is to enforce the equations of motion of the deformable body to follow the rigid body surface, unless separation occurs. The approach is utilized in the numerical simulation of pile installation in an undrained soil using an explicit large strain Lagrangean analysis. The analysis is focused on studying the effect of different pile tip shapes, values of adhesion factor and ratios of shear modulus to undrained shear strength in clays. Introduction The range of contact problems in geotechnical engineering covers many soilstructure interaction problems. Some of them involve large deformations and highly non-linear material behavior. Various problems concern with flexible structures, while in others the structure may be assumed to be rigid compared with the soil. In many cases, the shape of the contact surface between the soil and the structure is assumed to be linear, though in some cases the surface is non-linear. One way of solving contact problems is to use the finite element or finite difference methods. When deformations are large, these methods usually consider either the Lagrangean updating of the mesh or the Eularian stream of material through a fixed mesh. Strictly speaking, the Eulerian method is very suitable for steady motion and when field properties are fixed in space, while in the contrast cases it should include some assumptions, e.g. disregarding the evolution process of contact separation between the bodies. In these cases, the Lagrangean formulation may be suitable, though it might introduce new problems associated with highly distorted grid. Another problem that happens when using the Lagrangean formulation is encountered once the contact surface is highly non-linear. In which case, the grid points that travel along the contact surface may form geometrical discontinuities (or fictional tension stresses if a dimensionless nodal point element layer is used) due to an “arm effect” as described in figure 1a. This phenomenon is associated with contact formulations which do not allow overlapping of the deformable body grid lines and the rigid body non-linear surface. The phenomenon is accompanied by a non-smooth force distribution along interfaces (Hallquist et al. 1985). Figure 1 Typical picture of contact between deformable and rigid bodies. (a) using conventional interface, (b) using the new approach. This paper suggests a rigorous and physical approach that allows avoiding the geometrical discontinuities, which are caused due to the “arm effect”. At the current presentation we consider only the group of problems where one of the bodies is substantially rigid than the other, such that the contact surface could be represented by a traveling shape function. The basis of the approach is to impose the equation of motions on the deformable body grid points to follow a given shape function. Fictional gapping between the mesh and the shape function can still exist at concave surfaces. The “arm effect”, however, is disappeared, since the formulation allows for the deformable body grid lines to cross the shape function, while keeping the grid point (from which the strains are obtained) in continues contact with the rigid body surface. If contact logic is applied, the deformable body may separate from the rigid body and deform independently. Criteria for such contact logic may be for example conditioned on tensile strength. As mentioned previously, avoidance of the “arm effect” results with smooth stress distribution along the contact surface, and thus it is believed that the solution is more reliable. The applicability of the formulation to geotechnical engineering is demonstrated in the numerical simulation of pile installation where the tip is described by a nonlinear surface. An approach to contact between deformable and rigid bodies Rigid body motion A body may be defined as rigid if the distance between any two points of it is constant with time. The motion of a rigid body can always be described by only two components: a rotation tensor R and translation vector T. This can be formulated by a simple expression: T X R x ~ (1) where x and X are the current and initial position vectors, respectively, of a point on the rigid body in a fixed coordinate system, and the symbol “ ” denotes an inner product. Note that the translation vector T depends on the choice of the fixed origin, while the rotation tensor R does not. In order to relate eq. (1) to a function that describes the rigid body shape, a reference point to a moving local coordinate system is introduced by recognizing that eq. (1) holds for that point as for any other point on the rigid body. This implies that the reference point confirms to T X R x r r ~ , which upon subtraction from eq. (1) gives: l r X R x x ~ (2) where r l X X X is the position vector of a point on the rigid body in a local coordinates system. R may be evaluated by Euler angles if the rotation are infinitesimal. Otherwise, it may be evaluated using equations such as the Argyris form of the Euler-Rodrigues formula (Argyris, 1982), which are valid for arbitrarily large rotation angles. Similarly, the velocity transformation may be calculated using: