In this paper we prove a two-dimensional existence result for a variational model of crack growth for brittle materials in the realm of linearized elasticity. Starting with a time-discretized version of the evolution driven by a prescribed boundary load, we derive a time-continuous quasistatic crack growth in the framework of generalized special functions of bounded deformation (GSBD). As the time-discretization step tends to zero, the major difficulty lies in showing the stability of the static equilibrium condition, which is achieved by means of a Jump Transfer Lemma generalizing the result of [19] to the GSBD setting. Moreover, we present a general compactness theorem for this framework and prove existence of the evolution without imposing a-priori bounds on the displacements or applied body forces.

We derive Griffith functionals in the framework of linearized elasticity from nonlinear and frame indifferent energies in a brittle fracture via -convergence. The convergence is given in terms of rescaled displacement fields measuring the distance of deformations from piecewise rigid motions. The configurations of the limiting model consist of partitions of the material, corresponding piecewise rigid deformations and displacement fields which are defined separately on each component of the cracked body. Apart from the linearized Griffith energy the limiting functional also comprises the segmentation energy, which is necessary to disconnect the parts of the specimen.

We formulate a quasistatic nonlinear model for nonsimple viscoelastic materials at a finite-strain setting in the Kelvin's-Voigt's rheology where the viscosity stress tensor complies with the principle of time-continuous frame-indifference. We identify weak solutions in the nonlinear framework as limits of time-incremental problems for vanishing time increment. Moreover, we show that linearization around the identity leads to the standard system for linearized viscoelasticity and that solutions of the nonlinear system converge in a suitable sense to solutions of the linear one. The same property holds for time-discrete approximations and we provide a corresponding commutativity result. Our main tools are the theory of gradient flows in metric spaces and Γ-convergence.2010 Mathematics Subject Classification. 74D05, 74D10, 35A15, 35Q74, 49J45.

No abstract

No abstract

We establish a quantitative rigidity estimate for two-well frame-indifferent nonlinear energies, in the case in which the two wells have exactly one rank-one connection. Building upon this novel rigidity result, we then analyze solid-solid phase transitions in arbitrary space dimensions, under a suitable anisotropic penalization of second variations. By means of Γ-convergence, we show that, as the size of transition layers tends to zero, singularly perturbed two-well problems approach an effective sharp-interface model. The limiting energy is finite only for deformations which have the structure of a laminate. In this case, it is proportional to the total length of the interfaces between the two phases.

We present a piecewise Korn inequality for generalized special functions of bounded deformation (GSBD 2 ) in a planar setting generalizing the classical result in elasticity theory to the setting of functions with jump discontinuities. We show that for every configuration there is a partition of the domain such that on each component of the cracked body the distance of the function from an infinitesimal rigid motion can be controlled solely in terms of the linear elastic strain. In particular, the result implies that GSBD 2 functions have bounded variation after subtraction of a piecewise infinitesimal rigid motion. As an application we prove a density result in GSBD 2 in dimension two. Moreover, for all d ≥ 2 we show GSBD 2 (Ω) ⊂ (GBV (Ω; R)) d and the embedding SBD 2 (Ω) ∩ L ∞ (Ω; R d ) → SBV (Ω; R d ) into the space of special functions of bounded variation (SBV ). Finally, we present a Korn-Poincaré inequality for functions with small jump sets in arbitrary space dimension.2010 Mathematics Subject Classification. 74R10, 49J45, 70G75, 26D10.

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.