Nonlinear Elastic Fracture Research Articles

Variational Models with Eulerian–Lagrangian Formulation Allowing for Material Failure

We investigate the existence of minimizers of variational models featuring Eulerian–Lagrangian formulations. We consider energy functionals depending on the deformation of a body, defined on its reference configuration, and an Eulerian map defined on the unknown deformed configuration in the actual space. Our existence theory moves beyond the purely elastic setting and accounts for material failure by addressing free-discontinuity problems where both deformations and Eulerian fields are allowed to jump. To do this, we build upon the work of Henao and Mora-Corral regarding the variational modeling of cavitation and fracture in nonlinear elasticity. Two main settings are considered by modeling deformations as Sobolev and SBV-maps, respectively. The regularity of Eulerian maps is specified in each of these two settings according to the geometric and topological properties of the deformed configuration. We present some applications to specific models of liquid crystals, phase transitions, and ferromagnetic elastomers. Effectiveness and limitations of the theory are illustrated by means of explicit examples.

A numerical study of void coalescence and fracture in nonlinear elasticity

We present a numerical implementation of a model for void coalescence and fracture in nonlinear elasticity. The model is similar to the Ambrosio–Tortorelli regularization of the standard free-discontinuity variational model for quasistatic brittle fracture. The main change is the introduction of a nonlinear polyconvex energy that allows for cavitation. This change requires new analytic and numerical techniques. We propose a numerical method based on alternating directional minimization and a stabilized Crouzeix–Raviart finite element discretization. The method is used in several experiments, including void coalescence, void creation under tensile stress, failure in perfect materials and in materials with hard inclusions. The experimental results show the ability of the model and the numerical method to study different failure mechanisms in rubber-like materials.

Modeling of Externally Prestressed Beams until Fracture in Non Linear Elasticity

In this paper, we present an analytical model to analyze reinforced and prestressed concrete beams loaded in combined bending, axial load and shear, in the frame of non linear elasticity. In this model, the equilibrium of the beam is expressed by solving a system of equations, governing beams equilibrium, based on the stiffness matrix of the beam, which connects the load vector to the node displacements vector of the beam. It is built from the stiffness matrix of the section which takes into account a variation of the shearing modulus (depending on the shear variation) instead of assuming a constant shearing modulus as in linear elasticity. For the internal tendons, the stiffness matrix is completed by the terms due to the prestress effect in flexural equilibrium and by the balancing of one part of the shear by the transverse component of the force in the inclined cables.

Lusin's condition and the distributional determinant for deformations with finite energy

Based on a previous work by the authors on the modelling of cavitation and fracture in nonlinear elasticity, we give an alternative proof of a recent result by Csornyei, Hencl and Malý on the regularity of the inverse of homeomorphisms in the Sobolev space W 1;n 1 . With this aim, we show that the notion of fracture surface introduced by the authors in their model corresponds precisely to the original notion of cavity surface in the cavitation models of Muller and Spector (1995) and Conti and De Lellis (2003). We also find that the surface energy introduced in the model for cavitation and fracture is related to Lusin's condition (N) on the non-creation of matter. A fundamentalquestionunderlyingthis paperis whetherDetDu D detDunecessarily implies that the deformation u opens no cavities. We show that this is not true unless Muller and Spector's condition INV for the non-interpenetration of matter is satisfied. Having thus provided an additionaljustification of its importance, we provethe stability of this condition with respect to weak convergence in the critical spaceW 1;n 1 . Combining this with the work by Conti and De Lellis, we obtain an existence theory for cavitation in this critical case.

Vibration Analysis Method of Cracked Beam Based on the Principle of Energy

Using the theory of nonlinear elastic mechanics and fracture mechanics, the equation of motion governing equation of cracked beam is derived by the energy method, and solved with separation method of variables. Vibration analysis method based on the energy principle in this paper is proved feasible.Through numerical analysis, the effects of structural damping, crack location and depth on natural frequencies of linear vibration is investigated.

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.

Analytic parametric solutions for the HRR nonlinear elastic field with low hardening exponents

In this paper, we restore the already constructed approximate asymptotic solutions extracted in [10] concerning the HRR [1] strongly nonlinear fourth-order ordinary differential equation (ODE) for plane strain conditions in nonlinear elastic (plastic) fracture. It is proved that the above equation, for low strain hardening exponents (0 < N « 1), is reduced to a strongly nonlinear ODE of the second order. The method of the total differentials is used so that the last equation is reduced to Abels' equations of the second kind of the normal form, that can be analytically solved in parametric form. In addition, the case of rigid perfect-plasticity (N=0) is extensively investigated and several important results are extracted.

Simulation of crack growth in ductile materials

During crack growth of real materials, the total energy released can be partitioned into elastic and dissipative terms. By analyzing material models with mechanisms for dissipating energy and tracking all energy terms during crack growth, it is proposed that computer simulations of fracture can model crack growth by a total energy balance condition. One approach for developing fracture simulations is illustrated by analysis of elastic–plastic fracture. General equations were derived to predict crack growth and crack stability in terms of global energy release rate and irreversible energy effects. To distinguish plastic fracture from non-linear elastic fracture, it was necessary to imply an extra irreversible energy term. A key component of fracture simulations is to model this extra work. A model used here was to assume that the extra irreversible energy is proportional to the plastic work in a plastic-flow analysis. This idea was used to develop a virtual material based on Dugdale yield zones at the crack tips. A Dugdale virtual material was subjected to computer fracture experiments that showed it has many fracture properties in common with real ductile materials. A Dugdale material can serve as a model material for new simulations with the goal of studying the role of structure in the fracture properties of composites. One sample calculation showed that the toughness of a Dugdale material in an adhesive joint mimics the effect of joint thickness on the toughness of real adhesives. It is expected, however, that better virtual materials will be required before fracture simulations will be a viable approach to studying composite fracture. The approach of this paper is extensible to more advanced plasticity models and therefore to the development of better virtual materials.

On the Ubiquity of Fracture in Nonlinear Elasticity

This paper points out a subtle and little known consequence of assuming rank-one convexity for an isotropic hyperelastic material [e.g., of assuming a common ordering of principal stretches and principal stresses and hence the Baker–Ericksen inequalities]. What is shown is that rank-one convexity necessarily privilages those affine deformations which are dilatations: – the stored-energy associated with a dilatation is smaller than the stored energy associated with any other affine deformation possessing the same determinant. Also pointed out are fracture related consequences of this property that arise when the stored-energy function assigns to the dilatation of determinant δ > 0 a value A(δ∈) which is not an everywhere convex function of (0,∞).

A numerical method for detecting singular minimizers

A numerical method for computing minimizers in one-dimensional problems of the calculus of variations is described. Such minimizers may have unbounded derivatives, even when the integrand is smooth and regular. In such cases, because of the Lavrentiev phenomenon, standard finite element methods may fail to converge to a minimizer. The scheme proposed is shown to converge to an absolute minimizer and is tested on an example. The effect of quadrature is analyzed. The implications for higher-dimensional problems, and in particular for fracture in nonlinear elasticity, are discussed.