Crack on the boundary of two overlapping domains

Abstract

In this paper, we consider an overlapping domain problem for two elastic bodies. A glue condition of an equality-type is imposed at a given line. Simultaneously, a part of this line is considered to be a crack face with an inequality-type boundary condition describing mutual non-penetration between crack faces. Variational and differential formulations of the problem are considered. We prove a differentiability of the energy functional in the case of rectilinear cracks and find a formula for invariant integrals. Passage to the limit is justified provided that the rigidity of the body goes to infinity. (2000). 49J40 · 49K10 · 74R10. Mathematics Subject ClassificationKeywords. Crack with non-penetration · Derivative of energy functional · Overlapping domain. IntroductionIn solid mechanics, classical crack problems are characterized by linear boundary conditions imposed at the crack faces. It is well known that such linear models allow the opposite crack faces to penetrate each other which leads to inconsistency with practical situations. Since the beginning of 1990, the crack theory with non-penetration conditions has been under active study. This theory is characterized by inequality type boundary conditions at the crack faces. The book [6] contains results for crack models with nonpenetrations for different constitutive laws. In particular, 2D and 3D models as well as plate and shell models with inequality type boundary conditions are analyzed. After publication of this book, new approaches and trends in the study of non-linear crack models with non-penetration have been developed. For example, a problem of differentiation of the energy functional with respect to crack perturbations is solved in a general setting, smooth and fictitious domain methods are proposed, invariant integrals are constructed in cases of different geometrical situations, etc. We refer the reader to publications [5,[7][8][9]15,[17][18][19]. The latest results related to the crack growth for linear crack models (i.e. with linear boundary conditions at the crack faces) can be found in [2,4].There are other problems formulated in non-smooth domains with inequality type boundary conditions. It is known that contact problems for bodies of different dimensions are described by models similar to those of crack models with non-penetration. For these contact problems, the equilibrium equations are satisfied in cracked domains, and the boundary conditions are of inequality type. In fact, we should consider multi-domains with inequality type glue conditions (see [11,12]). As for models describing junctions of elastic bodies with equality type glue conditions we refer to [1] and [16] with the references quoted therein; elastic plate models can be found in [3]. Moreover, it turned out that many other practical problems should be described by crack models with overlapping domains. This overlapping approach is applicable for description of the subduction phenomenon of tectonic plates as well as for the slipping phenomenon of ice plates, and also for the ...

Junction problem for rigid and semirigid inclusions in elastic bodies

“…In addition to (8)-(9), we can also write junction conditions included in (6) and in the definition of K :…”

“…The results are concerned with a solution existence and qualitative properties of solutions. We can mention many other papers related to equilibrium problems with thin elastic and rigid inclusions and cracks; see [6,7,[10][11][12][13][14]30]. Optimal control problems for such models can be found in [15,[33][34][35].…”

On equilibrium problem for a two-layer structure in the presence of a defect

“…Предельный переход по параметру δ в задаче (10), (11) В этом пункте исследуется поведение решения задачи равновесия двуслойной упругой конструкции с дефектом в случаях, когда параметр повреждаемости дефекта δ стремится к предельным значениям, δ = 0 и δ = ∞. При каждом фиксированном δ ∈ (0, ∞) задача равновесия (10), (11) имеет вид: (20), (21) имеют место сходимости при δ → 0:…”

“…Теорема 3. Для последовательности решений (u δ , v δ ) семейства задач типа (20), (21) имеют место сходимости при δ → ∞:…”

“…В [16] рассмотрена задача равновесия для двуслойной конструкции, слои которой склеены на одном из берегов трещины, находящейся в нижнем слое, а вне линии склейки область контакта слоев заранее неизвестна; искомыми функциями в задаче являются как горизонтальные смещения, так и прогибы слоев. Задачи равновесия для различных двуслойных конструкций, формулируемые в плоской постановке, изучались в [17,18,19,20,21,22].…”