Linearization of quasistatic fracture evolution in brittle materials

The paper deals with the discontinuous Galerkin method (DGM) for the solution of compressible Navier-Stokes equations in the ALE form in time-dependent domains combined with the solution of linear and nonlinear dynamic elasticity. The developed methods are oriented to fluid-structure interaction (FSI), particularly to the simulation of air flow in a time-dependent domain representing vocal tract and vocal folds vibrations. We compare results obtained with the aid of linear and nonlinear elasticity models. The results show that it is more adequate to use the nonlinear elasticity St. Venant-Kirchhoff model in contrast to the linear elasticity model.

We study the atomistic-to-continuum limit for a model of quasi-static crack evolution driven by time-dependent boundary conditions. We consider a two-dimensional atomic mass spring system whose interactions are modeled by classical interaction potentials, supplemented by a suitable irreversibility condition accounting for the breaking of atomic bonding. In a simultaneous limit of vanishing interatomic distance and discretized time step, we identify a continuum model of quasistatic crack growth in brittle fracture [G. A. Francfort and J.-J. Marigo, Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids 46 (1998) 1319–1342] featuring an irreversibility condition, a global stability, and an energy balance. The proof of global stability relies on a careful adaptation of the jump-transfer argument in [G. A. Francfort and C. J. Larsen, Existence and convergence for quasi-static evolution in brittle fracture, Commun. Pure Appl. Math. 56 (2003) 1465–1500] to the atomistic setting.

Crack onset in stretched open hole PMMA plates considering linear and non-linear elastic behaviours

On an efficient octic order sub-parametric finite element method on curved domains

Surface penalization of self-interpenetration in linear and nonlinear elasticity

Crack propagation simulation in brittle elastic materials by a phase field method

Crack initiation in PMMA plates with circular holes considering kinetic energy and nonlinear elastic material behavior

\n We study the relations between a dynamic model proposed by Bourdin, Larsen and\n Richardson, and quasi-static fracture evolution. We assume the dynamic model has the\n boundary displacements of the material as input, and consider time-rescaled solutions of\n this model associated to a sequence of boundary conditions with speed going to zero. Next,\n we study whether this rescaled sequence converges to a function satisfying quasi-static\n fracture evolution. Under some hypotheses and assuming the speed of crack propagation\n slows down following the deceleration of boundary displacements, our main result shows\n that (up to a subsequence) the rescaled solutions converge to a quasi-static\n evolution.\n

Cracks, the major vehicle for material failure, tend to accelerate to high velocities in brittle materials. In three-dimensions, cracks generically undergo a micro-branching instability at about 40% of their sonic limiting velocity. Recent experiments showed that in sufficiently thin systems cracks unprecedentedly accelerate to nearly their limiting velocity without micro-branching, until they undergo an oscillatory instability. Despite their fundamental importance and apparent similarities to other instabilities in condensed-matter physics and materials science, these dynamic fracture instabilities remain poorly understood. They are not described by the classical theory of cracks, which assumes that linear elasticity is valid inside a stressed material and uses an extraneous local symmetry criterion to predict crack paths. Here we develop a model of two-dimensional dynamic brittle fracture capable of predicting arbitrary paths of ultra-high-speed cracks in the presence of elastic nonlinearity without extraneous criteria. We show, by extensive computations, that cracks undergo a dynamic oscillatory instability controlled by small-scale elastic nonlinearity near the crack tip. This instability occurs above a ultra-high critical velocity and features an intrinsic wavelength that increases proportionally to the ratio of the fracture energy to an elastic modulus, in quantitative agreement with experiments. This ratio emerges as a fundamental scaling length assumed to play no role in the classical theory of cracks, but shown here to strongly influence crack dynamics. Those results pave the way for resolving other long-standing puzzles in the failure of materials.

Shape stability of a gas cavity surrounded by linear and nonlinear elastic media

In this paper, we consider extensions of the gradient elasticity models proposed earlier by the second author to describe materials with fractional non-locality and fractality using the techniques developed recently by the first author. We derive a generalization of three-dimensional continuum gradient elasticity theory, starting from integral relations and assuming a weak non-locality of power-law type that gives constitutive relations with fractional Laplacian terms, by utilizing the fractional Taylor series in wave-vector space. In the sequel, we consider more general field equations with fractional derivatives of non-integer order to describe nonlinear elastic effects for gradient materials with power-law long-range interactions in the framework of weak non-locality approximation. The special constitutive relation that we elaborate upon can form the basis for developing a fractional extension of deformation theory of gradient plasticity. Using the perturbation method, we obtain corrections to the constitutive relations of linear fractional gradient elasticity, when the perturbations are caused by weak deviations from linear elasticity or by fractional gradient non-locality. Finally, we discuss fractal materials described by continuum models in non-integer dimensional spaces. Using a recently suggested vector calculus for non-integer dimensional spaces, we consider problems of fractal gradient elasticity.

This paper deals with the quasistatic crack growth of a homogeneous elastic brittle thin film. It is shown that the quasistatic evolution of a three-dimensional cylinder converges, as its thickness tends to zero, to a two-dimensional quasistatic evolution associated with the relaxed model. Firstly, a Γ-convergence analysis is performed with a surface energy density which does not provide weak compactness in the space of Special Functions of Bounded Variation. Then, the asymptotic analysis of the quasistatic crack evolution is presented in the case of bounded solutions that is with the simplifying assumption that every minimizing sequence is uniformly bounded in L∞.

The nonlinearly elastic Boussinesq problem uses the exact equations of finite elasticity to mathematically model the deformation produced in a homogeneous, isotropic, elastic half-space by a point force normal to the undeformed boundary. For this core problem of elasticity and engineering, the 1885 linear elasticity solution of Boussinesq is still used in a variety of applications despite the fact that the linear solution predicts physically unrealistic behavior in the primary region of interest beneath the load. In this paper, we aim to summarize two recent SIAM Journal of Applied Mathematics papers (D. A. Polignone Warne and P. G. Warne SIAM J. Appl. Math. 62 107-128 (2001) and SIAM J. Appl. Math. at press (2002)). These two papers develop asymptotic analyses for the nonlinearly elastic tensile point load problem for a half-space composed of either a general incompressible or compressible material, respectively. We also wish to present several new closed-form asymptotic solutions in the case of a tensile point load acting on an incompressible half-space. Finally, we comment on our approach to treating the complementary problem involving a compressive point load. Here we summarize the governing equations, conservation laws, hypotheses, and asymptotic tests needed to determine whether an isotropic hyperelastic material can support a finite deflection under a tensile point load. A variety of particular constitutive models in nonlinear elasticity are tested, and it is found that a material must be sufficiently stiff in order to support the load. Thus, it follows that many of the well-known strain-energy models for compressible hyperelastic materials proposed in the literature are unable to do so. For models which may sustain a tensile point load, we determine either the full asymptotic solution or the remaining equations and conditions for this solution. For classes of material models motivated by studies of Beatty and Jiang, we solve the resulting boundary value problem numerically in a compressible hyperelasticity setting, and we derive some new results including several closed-form explicit asymptotic solutions for a family of incompressible material models.

Elastography is a noninvasive imaging technique that provides information on soft tissue stiffness. Young's modulus is typically used to characterize soft tissues' response to the applied force, as soft tissues are often considered linear elastic, isotropic, and quasi-incompressible materials. This approximation is reasonable for small strains, but soft tissues undergo large deformations also for small values of force and exhibit nonlinear elastic behavior. Outside the linear regime, the elastic modulus is dependent on the strain level and is different for any kind of tissue. The aim of this study was to characterize, ex vivo, the mechanical response of two different mice muscles to an external force. A system for transverse force-controlled uniaxial compression enabled obtaining the stress-strain (σ-ε) curve of the samples. The strain-dependent Young's modulus (SYM) model was adopted to reproduce muscle compression behavior and to predict the elastic modulus for large deformations. After that, a recursive linear model was employed to identify the initial linear region of the σ-ε curve. Results showed that both muscle types exhibited a strain hardening effect and that the SYM model provided good fitting of the entire σ-ε curves. The application of the recursive linear model allowed capturing the initial linear region in which the approximation of these tissues as linear elastic materials is reasonable. The residual analysis displayed that even if the SYM model better summarizes the muscle behavior on the entire region, the linear model is more precise when considering only the initial part of the σ-ε curve.

Stable Numerical Schemes for Fluids, Structures and their Interactions