Non-interpenetration Conditions Research Articles

Non-interpenetration conditions in the passage from nonlinear to linearized Griffith fracture

We characterize the passage from nonlinear to linearized Griffith-fracture theories under non-interpenetration constraints. In particular, sequences of deformations satisfying a Ciarlet-Nečas condition in SBV2 and for which a convergence of the energies is ensured, are shown to admit asymptotic representations in GSBD2 satisfying a suitable contact condition. With an explicit counterexample, we prove that this result fails if convergence of the energies does not hold. We further prove that each limiting displacement satisfying the contact condition can be approximated by an energy-convergent sequence of deformations fulfilling a Ciarlet-Nečas condition. The proof relies on a piecewise Korn-Poincaré inequality in GSBD2, on a careful blow-up analysis around jump points, as well as on a refined GSBD2-density result guaranteeing enhanced contact conditions for the approximants.

An accurate local average contact method for nonmatching meshes

The present paper deals with linear and quadratic finite element approximations of the two and three-dimensional unilateral contact problems between two elastic bodies with nonmatching meshes. We propose a simple noninterpenetration condition on the displacements which is local as the well known node-to-segment and node-to-face conditions and accurate like the mortar approach. This condition consists of averaging locally on a few elements the noninterpenetration. We prove optimal convergence rates in 2D and 3D using various linear and quadratic elements. The Taylor patch test and the Hertzian contact test illustrate the theoretical results and show the capabilities of the method.

Phase-Field Modeling of Fracture in Ferroelectric Materials

This paper presents a family of phase-field models for the coupled simulation of the microstructure formation and evolution, and the nucleation and propagation of cracks in single and polycrystalline ferroelectric materials. The first objective is to introduce a phase-field model for ferroelectric single crystals. The model naturally couples two existing energetic phase-field approaches for brittle fracture and ferroelectric domain formation and evolution. Simulations show the interactions between the microstructure and the crack under mechanical and electromechanical loadings. Another objective of this paper is to encode different crack face boundary conditions into the phase-field framework since these conditions strongly affect the fracture behavior of ferroelectrics. The smeared imposition of these conditions are discussed and the results are compared with that of sharp crack models to validate the proposed approaches. Simulations show the effects of different conditions and electromechanical loadings on the crack propagation. In a third step, the model is modified by introducing a crack non-interpenetration condition in the variational approach to fracture accounting for the asymmetric behavior in tension and compression. The modified model makes it possible to explain anisotropic crack growth in ferroelectrics under the Vickers indentation loading. This model is also employed for the fracture analysis of multilayer ferroelectric actuators, which shows the potential of the model for future applications. The coupled phase-field model is also extended to polycrystals by introducing realistic polycrystalline microstructures in the model. Inter- and trans-granular crack propagation modes are observed in the simulations. Finally, and for completeness, the phase-field theory is extended to the simulation of the propagation of conducting cracks under purely electrical loading and to the three-dimensional simulation of crack propagation in ferroelectric single crystals. Salient features of the crack propagation phenomenon predicted by the simulations of this paper are directly compared with experimental observations.

Analysis of micro fracture in human Haversian cortical bone under compression

A procedure to investigate local stress intensity factors in human Haversian cortical bone under compression is presented. The method combines a customised experimental setting for micro-compression tests of millimetric bone specimens and a finite element contact model conforming to the bone morphology that tracks advancing microcracks. The non-interpenetration conditions along the crack edges are ensured by penalty constraints of which the parameters are optimised for minimum contact pressure error with respect to the crack orientations. A cohesive crack opening law is implemented in the wake of the crack tips to remain consistent with the progressive tearing of collagen fibrils. The displacement solution is searched by a Newton-Raphson scheme containing a double loop first on the displacements and second on the frictional contact and cohesive condition updates at the crack interfaces. The experimental Dirichlet boundary conditions are acquired by digital image cross-correlation of bone light microscopy observations and then imported into the model. The local mechanical elastic moduli are measured by nanoindentation and microextensometry. The comparison of the macroscopic stress-strain numerical response with the experiment reveals the existence of narrow diffuse damaged zones near the major cracks where the local stress intensity factors can be calculated.

Multi-layered folding with voids

In the deformation of layered materials such as geological strata, or stacks of paper, mechanical properties compete with the geometry of layering. Smooth, rounded corners lead to voids between the layers, while close packing of the layers results in geometrically induced curvature singularities. When voids are penalized by external pressure, the system is forced to trade off these competing effects, leading to sometimes striking periodic patterns. In this paper, we construct a simple model of geometrically nonlinear multi-layered structures under axial loading and pressure confinement, with non-interpenetration conditions separating the layers. Energy minimizers are characterized as solutions of a set of fourth-order nonlinear differential equations with contact-force Lagrange multipliers, or equivalently of a fourth-order free-boundary problem. We numerically investigate the solutions of this free-boundary problem and compare them with the periodic solutions observed experimentally.

Invertibility and Weak Continuity of the Determinant for the Modelling of Cavitation and Fracture in Nonlinear Elasticity

In this paper, we present and analyze a variational model in nonlinear elasticity that allows for cavitation and fracture. The main idea in unifying the theories of cavitation and fracture is to regard both cavities and cracks as phenomena of the creation of a new surface. Accordingly, we define a functional that measures the area of the created surface. This functional has relationships with the theory of Cartesian currents. We show that the boundedness of that functional implies sequential weak continuity of the determinant of the deformation gradient, and that the weak limit of one-to-one almost everywhere deformations is also one-to-one almost everywhere. We then use these results to obtain the existence of minimizers of variational models that incorporate elastic energy and this created surface energy, taking into account orientation-preserving and non-interpenetration conditions.

Quasistatic crack growth in finite elasticity with non-interpenetration

We present a variational model to study the quasistatic growth of brittle cracks in hyperelastic materials, in the framework of finite elasticity, taking into account the non-interpenetration condition.

Regularized formulation of the variational brittle fracture with unilateral contact: Numerical experiments

This paper presents a modified regularized formulation of the Ambrosio–Tortorelli type to introduce the crack non-interpenetration condition in the variational approach to fracture mechanics proposed by Francfort and Marigo [1998. Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46 (8), 1319–1342]. We focus on the linear elastic case where the contact condition appears as a local unilateral constraint on the displacement jump at the crack surfaces. The regularized model is obtained by splitting the strain energy in a spherical and a deviatoric parts and accounting for the sign of the local volume change. The numerical implementation is based on a standard finite element discretization and on the adaptation of an alternate minimization algorithm used in previous works. The new regularization avoids crack interpenetration and predicts asymmetric results in traction and in compression. Even though we do not exhibit any gamma-convergence proof toward the desired limit behavior, we illustrate through several numerical case studies the pertinence of the new model in comparison to other approaches.

Minimal energy configurations of strained multi-layers

We derive an effective plate theory for internally stressed thin elastic layers as are used, e.g., in the fabrication of nano- and microscrolls. The shape of the energy minimizers of the effective energy functional is investigated without a priori assumptions on the geometry. For configurations in two dimensions (corresponding to Euler-Bernoulli theory) we also take into account a non-interpenetration condition for films of small but non-vanishing thickness.

Quadratic finite element methods for unilateral contact problems

The present paper is concerned with the frictionless unilateral contact problem between two elastic bodies in a bidimensional context. We consider a mixed formulation in which the unknowns are the displacement field and the contact pressure. We introduce a finite element method using quadratic elements and continuous piecewise quadratic multipliers on the contact zone. The discrete unilateral non-interpenetration condition is either an exact non-interpenetration condition or only a nodal condition. In both cases, we study the convergence of the finite element solutions and a priori error estimates are given. Finally, we perform the numerical comparison of the quadratic approach with linear finite elements.

Fracture growth-I. Formulation and implementations

SUMMARY The orientation of fracture growth 0 can be determined by using the maximum strain-energy release-rate criterion G,,,. When a fracture is under compression, the evaluation of G(8) involves the solution to frictional contact problems, where the fracture faces are constrained by the non-interpenetration condition and a friction law. We propose a repulsion scheme to handle these constraints on the fracture faces: the interpenetration is iteratively eliminated by adjusting the normal compressive force (repulsion), and the friction law is satisfied by modifying the friction resistance at each iteration. Under uniaxial compression, the numerical results show that an artificial cut extends by a kink, which is consistent with laboratory observations. Fractures under pure mode I1 (simple shearing) and under mixed-mode loading (transtension) grow by a kink and along a smooth, slightly convex trajectory; the computed path is almost identical to the one obtained in the laboratory.

Contour design for contact stress minimization by interior penalty method

The present study has exploited optimization techniques in the framework of the automated direct method to design optimal shapes of contact surfaces. The developed model consists of two nonlinear submodels; the first is concerned with the design problem in which the peak contact pressure is taken as an objective function to be minimized subject to a set of constraints representing the limitations on contact surface dimensions and the maximum effective stresses. The peak contact pressure is an implicit function of the design variables and state variables. Once the contact problem of two bodies of specific surfaces dimensions (design variables) is solved, the peak contact pressure can be computed directly as a qualifying function of the state variables (nodal displacements). The other submodel represents the elastostatic contact problem formulated as a nonlinear mathematical one in which the potential energy is taken as the objective function to be minimized for equilibrium state and the relevant constraints represent the noninterpenetration conditions. The proposed algorithm adopts the interior penalty method enhanced by the direct automated procedure to obtain the optimal surfaces dimensions. Two examples of different nature are presented for illustration.

Finite element analysis of elastoplastic contact problems with friction

Fang, T. and Wang, Z. N., A Generalization of Caughey's Normal Mode Approach to Nonlinear Random Vibration Problems, to be published in AIAA Journal. Caughey, T. K., Equivalent Linearization Techniques, Journal of the Acoustical Society of America, Vol. 35, No. 11, 1963, pp. 1706-1711. that is taken to establish the non-interpenetration conditions between components D and W. The variables X$ X^^ represent the Cartesian coordinates of the material points in contact components D and W. The functional EM can be expressed as

Frictionless contact of elastic bodies by finite element method and mathematical programming technique

The work presents an analysis of the contact problems between elastic bodies without friction. Extended variational principles in the contact problems constitute the theoretical foundations of the analysis. In fact, kinematically admissible displacement fields are subjected to the non-interpenetration conditions so that the contact problem being equivalent to a minimization of the potential energy with contraints on the displacements may be reduced to a particular case of the mathematical programming techniques. Applications are limited to plane state of stress or strain cases where small deformations are considered. The geometrical space is discretized into finite elements, namely the 12 degree of freedom hybrid triangles or the 16 degrees of freedom isoparametric bi-linear quadrangle elements. A special procedure of linearization is adopted to use the quadratic programming algorithm already prepared for large scale problems. In these conditions, a computer program has been written in FORTRAN IV H and the results obtained from this present good agreement comparing to analytical solutions or other numerical solution in the literature. Finally, theoretical efforts have been performed to generalize the method by introducing a concept of fictitious locked material which represents the assumed contact space between the two bodies. A general and new condition of non-interpenetration is proposed for general cases of two elastic solids in contact.