Residual stresses caused by linear incompatibility of strains in an orthotropic cylindrical shell

We analyze a healed crack in an orthotropic cylindrical nonshallow shell using a continuously distributed dislocation model. The field of residual stresses on the faces of the shell is investigated for different cases of transverse dislocations of normal opening.     

Energy release rate for cracks in finite‐strain elasticity

Griffith's fracture criterion describes in a quasistatic setting whether or not a pre‐existing crack in an elastic body is stationary for given external forces. In terms of the energy release rate (ERR), which is the derivative of the deformation energy of the body with respect to a virtual crack extension, this criterion reads: if the ERR is less than a specific constant, then the crack is stationary, otherwise it will grow. In this paper, we consider geometrically nonlinear elastic models with polyconvex energy densities and prove that the ERR is well defined. Moreover, without making any assumption on the smoothness of minimizers, we rigorously derive the well‐known Griffith formula and the J‐integral, from which the ERR can be calculated. The proofs are based on a weak convergence result for Eshelby tensors. Copyright © 2007 John Wiley & Sons, Ltd.     

A crack model with delayed embedded discontinuities

In this contribution the combination of a smeared rotating crack model with a crack model based on the strong discontinuity approach and formulated within the framework of elements with embedded discontinuities is presented. This so–called crack model with delayed embedded discontinuities allows considering crack opening in the direction normal to the crack and relative tangential displacements of the crack faces with transfer of shear forces across the crack faces. The advantages of this approach are shown by the numerical simulation of an anchor pull out test. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Local minimizers with vortices to the Ginzburg‐Landau system in three dimensions

We construct local minimizers to the Ginzburg‐Landau energy in certain three‐dimensional domains based on the asymptotic connection between the energy and the total length of vortices using the theory of weak Jacobians. Whenever there exists a collection of locally minimal line segments spanning the domain, we can find local minimizers with arbitrarily assigned degrees with respect to each segment. © 2003 Wiley Periodicals, Inc.     

A Crack with Interfacial Bonds in an Isotropic Medium Strengthened with a Regular System of Stringers

An isotropic medium containing a system of foreign transverse rectilinear inclusions is considered. Such a medium can be interpreted as an infinite plate strengthened with a regular system of ribs (stringers) whose cross section is a very narrow rectangle. The medium is weakened by a rectilinear crack. The action of the stringers is replaced with unknown equivalent concentrated forces at the points of their connection with the medium. A model of a crack with areas where its faces interact with each other is investigated. This interaction is modeled by introducing bonds (adhesion forces) between faces in the crack tip zone. The boundary-value problem on equilibrium of the crack under the action of external tensile forces is reduced to a nonlinear singular integral equation, from the solution of which the tractions in the bonds are found. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 41, No. 6, pp. 773–782, November–December, 2005.     

Axisymmetrical vibrations of shells of revolution under abrupt loading

A frequency method is proposed for solving the problem of the vibrations of shells of revolution taking into account the energy dissipation under arbitrary force loading and on collision with a rigid obstacle. The Laplace transform is taken of the equation of the vibrations of a shell of revolution with non-zero initial conditions. For the inhomogeneous differential equation obtained, a variational method is used to solve the boundary-value problem, which consists of finding the Laplace-transformed boundary transverse and longitudinal forces and bending moments as functions of the boundary displacements. The equations of equilibrium of nodes, i.e. the corresponding equations of the finite-element method, are then compared, using results obtained earlier [1–4]. Amplitude-phase-frequency characteristics (APFCs) for the shell cross-sections selected are plotted. An inverse Laplace transformation is carried out using the clear relationship between the extreme points of the APFCs and the coefficients of the corresponding terms of the series in an expansion vibration modes [3]. In view of the fact that the proposed approach is approximate, numerical testing is used.     

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.      

On the problem of axial tension of a circular inhomogeneous transversally isotropic shell

We consider the problem of axial tension of a circular inhomogeneous cylindrical shell by longitudinal forces. The determination of the stress-strain state of the shell is performed on the basis of a refined theory. The possibility of losing local stability under axial tension is discussed.     

Nonsmooth domain optimization for elliptic equations with unilateral conditions

In the paper we consider elliptic boundary problems in domains having cuts (cracks). The non-penetration condition of inequality type is prescribed at the crack faces. A dependence of the derivative of the energy functional with respect to variations of crack shape is investigated. This shape derivative can be associated with the crack propagation criterion in the elasticity theory. We analyze an optimization problem of finding the crack shape which provides a minimum of the energy functional derivative with respect to a perturbation parameter and prove a solution existence to this problem.     

From phase field to sharp cracks: Convergence of quasi-static evolutions in a special setting

We consider the quasi-static evolution of a straight crack within the recently developed phase-field approach and the classical sharp crack approach, and we show a strong correlation between the outcomes from the two approaches: the corresponding energies, minimizers, energy release rates and quasi-static evolutions converge as the internal length parameter of the phase-field model tends to zero. A crucial point in the proof is a novel representation of the energy release rate, which allows one to pass to the limit under weak convergence of the strains.     

Delamination buckling of a rectangular orthotropic composite plate containing a band crack

The delamination buckling problem for a rectangular plate made of an orthotropic composite material is studied. The plate contains a band crack whose faces have an initial infinitesimal imperfection. The subsequent development of this imperfection due to an external compressive load acting along the crack is studied through the use of the three-dimensional geometrically nonlinear field equations of elasticity theory for anisotropic bodies. A criterion of initial imperfection is used in determining the critical forces. The corresponding boundary-value problems are solved by employing the boundary-form perturbation technique and the FEM. Numerical results for the critical force are presented.     

Solution of dynamic problems in crack theory for large times

A technique is proposed for the solution of three-dimensional dynamic problems in the mathematical theory of cracks, at times larger than a characteristic value, which depends on the geometric parameters of the crack, and on the propagation velocities of longitudinal and transverse waves in the body. The stress-intensity factors are determined as functions of time, for specific types of external loading on an infinite body with a disc-shaped crack.Translated from Matematicheskie Metody i Fiziko-mekhanicheskie Polya, No. 26, pp. 24–29, 1987.     

Analysis of the stress-strain state of a shell structure with a large cutout

The finite-element method is applied to analyze the stress-strain state of a composite structure consisting of a shell supported by longitudinal and transverse beams with an internal deck also supported by longitudinal and transverse beams. The analysis reveals the presence of stress concentrations near the cutout, at the application points of concentrated forces, and near the inclined frames.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 59, pp. 67–70, 1986.     

Influence of Initial Stresses on Stress Intensity Factors at Crack Tips in a Composite Strip

The influence of initial tension or compression along cracks on the stress intensity factor (SIF) at crack tips under the action of additional normal forces on crack edges is studied for infinite bodies. A strip made of a composite material is considered. The strip ends are simply supported, and the strip contains a crack whose edges are parallel to its face planes. The strip is first stretched or compressed along crack edges, and then additional uniformly distributed normal forces are applied to the crack edges. The influence of the initial tension (compression) on the SIF caused by the additional normal forces is studied. The corresponding boundary-value problems are modelled with the use of the three-dimensional linearized theory of elasticity. All the investigations are carried out numerically by employing the finite-element method. The values of SIF are calculated by the energy release method.     

Implementation of a cohesive zone model for the investigation of the dynamic behavior of a rotating shaft with a transverse crack

The influence of a transverse crack on the vibration of a rotating shaft has been at the focus of attention of many researchers. The knowledge of the dynamic behavior of cracked shaft has helped in predicting the presence of a crack in a rotor. Here, the changing stiffness of the cracked shaft is investigated based on a cohesive zone model. This model is developed for mode-I plane strain and accounts for triaxiality of the stress state explicitly by using basic elastic-plastic constitutive relations. Then, the proposed numerical solution is compared to the switching crack model, which is based on linear elastic fracture mechanics. The cohesive zone model is implemented in finite element techniques to predict and to analyse the dynamic behavior of cracked rotor system. Timoshenko beam theory is used to model the discrete shaft under the effect of gravity, unbalance force and gyroscopic effect. The analysis includes the cohesive function for describing the breathing crack and the reduction of the second moment of area of the element at the location of the crack. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Epsilon‐stable quasi‐static brittle fracture evolution

We introduce a new definition of stability, ε‐stability, that implies local minimality and is robust enough for passing from discrete‐time to continuous‐time quasi‐static evolutions, even with very irregular energies. We use this to give the first existence result for quasi‐static crack evolutions that both predicts crack paths and produces states that are local minimizers at every time, but not necessarily global minimizers. The key ingredient in our model is the physically reasonable property, absent in global minimization models, that whenever there is a jump in time from one state to another, there must be a continuous path from the earlier state to the later along which the energy is almost decreasing. It follows that these evolutions are much closer to satisfying Griffith's criterion for crack growth than are solutions based on global minimization, and initiation is more physical than in global minimization models. © 2009 Wiley Periodicals, Inc.     

Primal Dual Methods for Wasserstein Gradient Flows

Combining the classical theory of optimal transport with modern operator splitting techniques, we develop a new numerical method for nonlinear, nonlocal partial differential equations, arising in models of porous media, materials science, and biological swarming. Our method proceeds as follows: first, we discretize in time, either via the classical JKO scheme or via a novel Crank–Nicolson-type method we introduce. Next, we use the Benamou–Brenier dynamical characterization of the Wasserstein distance to reduce computing the solution of the discrete time equations to solving fully discrete minimization problems, with strictly convex objective functions and linear constraints. Third, we compute the minimizers by applying a recently introduced, provably convergent primal dual splitting scheme for three operators (Yan in J Sci Comput 1–20, 2018). By leveraging the PDEs’ underlying variational structure, our method overcomes stability issues present in previous numerical work built on explicit time discretizations, which suffer due to the equations’ strong nonlinearities and degeneracies. Our method is also naturally positivity and mass preserving and, in the case of the JKO scheme, energy decreasing. We prove that minimizers of the fully discrete problem converge to minimizers of the spatially continuous, discrete time problem as the spatial discretization is refined. We conclude with simulations of nonlinear PDEs and Wasserstein geodesics in one and two dimensions that illustrate the key properties of our approach, including higher-order convergence our novel Crank–Nicolson-type method, when compared to the classical JKO method.

     

An Atomistic-to-Continuum Analysis of Crystal Cleavage in a Two-Dimensional Model Problem

A two-dimensional atomic mass spring system is investigated for critical fracture loads and its crack path geometry. We rigorously prove that, in the discrete-to-continuum limit, the minimal energy of a crystal under uniaxial tension leads to a universal cleavage law and energy minimizers are either homogeneous elastic deformations or configurations that are completely cracked and do not store elastic energy. Beyond critical loading, the specimen generically cleaves along a unique optimal crystallographic hyperplane. For specific symmetric crystal orientations, however, cleavage might fail. In this case a complete characterization of possible limiting crack geometries is obtained.     

Aggregation-diffusion to constrained interaction: Minimizers & gradient flows in the slow diffusion limit

Inspired by recent work on minimizers and gradient flows of constrained interaction energies, we prove that these energies arise as the slow diffusion limit of well-known aggregation-diffusion energies. We show that minimizers of aggregation-diffusion energies converge to a minimizer of the constrained interaction energy and gradient flows converge to a gradient flow. Our results apply to a range of interaction potentials, including singular attractive and repulsive-attractive power-law potentials. In the process of obtaining the slow diffusion limit, we also extend the well-posedness theory for aggregation-diffusion equations and Wasserstein gradient flows to admit a wide range of nonconvex interaction potentials. We conclude by applying our results to develop a numerical method for constrained interaction energies, which we use to investigate open questions on set valued minimizers.     

Shape optimization of rigid inclusions for elastic plates with cracks

In the paper, we consider an optimal control problem of finding the most safe rigid inclusion shapes in elastic plates with cracks from the viewpoint of the Griffith rupture criterion. We make use of a general Kirchhoff–Love plate model with both vertical and horizontal displacements, and nonpenetration conditions are fulfilled on the crack faces. The dependence of the first derivative of the energy functional with respect to the crack length on regular shape perturbations of the rigid inclusion is analyzed. It is shown that there exists a solution of the optimal control problem.