| 1. |
- Abali, Bilen Emek, et al.
(författare)
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A novel phase‐field approach to brittle damage mechanics of gradient metamaterials combining action formalism and history variable
- 2021
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Ingår i: Zeitschrift für angewandte Mathematik und Mechanik. - : John Wiley & Sons. - 0044-2267 .- 1521-4001. ; 101:9
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Tidskriftsartikel (refereegranskat)abstract
- Metamaterials response is generally modeled by generalized continuum based theories. Their inherent substructure leads to a necessity for higher-order theories, and especially in damage mechanics, such a generalization is difficult to acquire. We exploit the action formalism in order to obtain the governing equations in generalized damage mechanics for metamaterials. Additionally, by using auxilliary variables, the variational formulation is endowed with the first rate of damage variable that is missing in standard approaches. The presented action formalism with auxilliary variables leads directly to the weak form. We implement a finite element method based approach by using open-source computing platform called FEniCS and solve this weak in order to obtain the deformation and damage numerically. Metamaterials simulations are demonstrated for simple geometries in mixed mode (I and II) as well as in mode III.
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| 2. |
- Abali, Bilen Emek
(författare)
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Energy based methods applied in mechanics by using the extended Noether's formalism
- 2023
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Ingår i: Zeitschrift für angewandte Mathematik und Mechanik. - : Wiley-VCH Verlagsgesellschaft. - 0044-2267 .- 1521-4001. ; 103:12
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Tidskriftsartikel (refereegranskat)abstract
- Physical systems are modeled by field equations; these are coupled, partial differential equations in space and time. Field equations are often given by balance equations and constitutive equations, where the former are axiomatically given and the latter are thermodynamically derived. This approach is useful in thermomechanics and electromagnetism, yet challenges arise once we apply it in damage mechanics for generalized continua. For deriving governing equations, an alternative method is based on a variational framework known as the extended Noether's formalism. Its formal introduction relies on mathematical concepts limiting its use in applied mechanics as a field theory. In this work, we demonstrate the power of extended Noether's formalism by using tensor algebra and usual continuum mechanics nomenclature. We demonstrate derivation of field equations in damage mechanics for generalized continua, specifically in the case of strain gradient elasticity.
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| 3. |
- Aiki, Toyohiko, et al.
(författare)
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A free boundary problem describing migration into rubbers : Quest of the large time behavior
- 2022
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Ingår i: Zeitschrift für angewandte Mathematik und Mechanik. - : John Wiley & Sons. - 0044-2267 .- 1521-4001. ; 102:7
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Tidskriftsartikel (refereegranskat)abstract
- In many industrial applications, rubber-based materials are routinely used in conjunction with various penetrants or diluents in gaseous or liquid form. It is of interest to estimate theoretically the penetration depth as well as the amount of diffusants stored inside the material. In this framework, we prove the global solvability and explore the large time-behavior of solutions to a one-phase free boundary problem with nonlinear kinetic condition that is able to describe the migration of diffusants into rubber. The key idea in the proof of the large time behavior is to benefit from a contradiction argument, since it is difficult to obtain uniform estimates for the growth rate of the free boundary due to the use of a Robin boundary condition posed at the fixed boundary.
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| 4. |
- Andersson, Lars-Erik, et al.
(författare)
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Existence and uniqueness for frictional incremental and rate problems - sharp critical bounds
- 2016
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Ingår i: Zeitschrift für angewandte Mathematik und Mechanik. - : WILEY-V C H VERLAG GMBH. - 0044-2267 .- 1521-4001. ; 96:1, s. 78-105
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Tidskriftsartikel (refereegranskat)abstract
- We investigate frictional contact problems for discrete linear elastic structures, in particular the quasistatic incremental problem and the rate problem. It is shown that sharp conditions on the coefficients of friction for unique solvability of these problems are the same. We also give explicit expressions of these critical bounds by using a method of optimization. For the case of two spatial dimensions the conditions are formulated as a huge set of non symmetric eigenvalue problem. A computer program for solving these problems was designed and used to compute the critical bounds for some structures of relative small size, some of which appeared in the literature. The results of a variety of numerical experiments with uniform and non uniform distributions of the frictional properties are presented. (C) 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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| 5. |
- Arévalo, Carmen, et al.
(författare)
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β-blocked multistep methods for Euler-Lagrange DAEs: Linear analysis
- 1997
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Ingår i: ZAMM Zeitschrift fur Angewandte Mathematik und Mechanik. - : Wiley. - 0044-2267. ; 77:8, s. 609-617
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Tidskriftsartikel (refereegranskat)abstract
- Many different methods have been suggested for the numerical solution of index 2 and 3 Euler-Lagrange equations. We focus on 0-stability of multistep methods (φ, σ) and investigate the relations between some well-known computational techniques. By various modifications, referred to as β-blocking of the σ polynomial, some basic shortcomings of multistep methods may be overcome. This approach is related to projection techniques and has a clear and well-known analogy in control theory. In particular, it is not necessary to use BDF methods for the solution of high index problems; indeed, “nonstiff” methods may be used for part of the system provided that the state-space form is nonstiff. We illustrate the techniques and demonstrate the results with a simplified multibody model of a truck.
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| 6. |
- Bader, Thomas K., et al.
(författare)
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Poromechanical scale transitions of failure stresses in wood : from the lignin to the spruce level
- 2010
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Ingår i: Zeitschrift für angewandte Mathematik und Mechanik. - : Wiley. - 0044-2267 .- 1521-4001. ; 90:10-11, s. 750-767
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Tidskriftsartikel (refereegranskat)abstract
- Wood strength is highly anisotropic, due to the inherent structural hierarchy of the material. In the framework of a combined random-periodic multiscale poro-micromechanics model, we here translate compositional information throughout this hierarchy into the resulting anisotropic strength at the softwood level, based on “universal” elastic properties of cellulose, hemicellulose, and lignin, and on the shear strength of the latter elementary constituent. Relating, through elastic energy-derived higher-order strains in a poromechanical representative volume element, the (quasi-)brittle failure of lignin to overall softwood failure, results in a macroscopic microstructure-dependent failure criterion for softwood. The latter satisfactorily predicts the biaxial strength of spruce at various loading angles with respect to the grain direction. The model also predicts the experimentally well-established fact that uniaxial tensile and compressive strengths, as well as the shear strength of softwood, depend quasi-linearly on the cell water content, but highly nonlinearly on the lumen porosity.
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| 7. |
- Burdakov, Oleg, 1953-
(författare)
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On Superlinear Convergence of Some Stable Variants of the Secant Method
- 1986
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Ingår i: Zeitschrift für angewandte Mathematik und Mechanik. - : Wiley. - 0044-2267 .- 1521-4001. ; 66:2, s. 615-622
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Tidskriftsartikel (refereegranskat)abstract
- In a recent author's paper four classes of methods of the secant type for solving systems of nonlinear equations were proposed. They are stable with respect to linear dependence of the search directions. If the Jacobian matrix is symmetric, then two of them take into account this property. It is proved here that some four special methods from the classes converge superlinearly.
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| 8. |
- Burdakov, Oleg, 1953-
(författare)
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On using the minimum spanning tree algorithm for optimal secant approximation of derivatives
- 1996
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Ingår i: Zeitschrift für angewandte Mathematik und Mechanik. - : Wiley. - 0044-2267 .- 1521-4001. ; 76:S3, s. 389-390
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Tidskriftsartikel (refereegranskat)abstract
- The following problem is considered. Given m + 1 points {xi}0m in Rn which generate an m-dimensional linear manifold, construct for this manifold a maximally linearly independent basis that consists of vectors of the form xi - xj. This problem is present in, e.g., stable variants of the secant method, where it is required to approximate the Jacobian matrix f' of a nonlinear mapping f by using values off computed at m + 1 points. In this case, it is also desirable to have a combination of finite differences with maximal linear independence. As a natural measure of linear independence, we consider a functional which is maximized to find an optimal combination of m pairs {xi, xj}. We show that the problem is not of combinatorial complexity but can be reduced to the minimum spanning tree (MST) problem, which is solved by an MST-type algorithm in O(m2n) time.
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| 9. |
- Chandler-Wilde, S. N., et al.
(författare)
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A boundary integral equation formulation for the Helmholtz equation in a locally perturbed half-plane
- 2005
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Ingår i: Zeitschrift für angewandte Mathematik und Mechanik. - : Wiley. - 0044-2267 .- 1521-4001. ; 85:2, s. 79-88
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we study the application of boundary integral equation methods to the solution of the Helmholtz equation in a locally perturbed half-plane with Robin or impedance boundary conditions. This problem models outdoor noise propagation from a cutting onto a surrounding flat plane, and also the harbour resonance problem in coastal engineering. We employ Green's theorem to derive a system of three coupled integral equations. The three unknowns are the pressure on the boundary of the disturbance and the pressure and its normal derivative on the interface with the upper half-space. We prove that the integral equation formulation has a unique solution at all wavenumbers by proving equivalence of the boundary value problem and the integral equation formulation and proving uniqueness of solution for the boundary value problem.
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| 10. |
- Diaz, José, et al.
(författare)
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A wavelet semidiscretisation of elastic multibody systems
- 2003
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Ingår i: Zeitschrift für Angewandte Mathematik und Mechanik. - : Wiley. - 0044-2267. ; 83:10, s. 677-689
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we consider a coupled system of elastic and rigid bodies. It can be mathematically formulated as a coupled system of ordinary and partial differential equations, often written in its weak form. The discretisation is performed by a Galerkin-type ansatz in connection with a finite element approach or known eigenfunctions. Here, we demonstrate instead the use of a recently published Galerkin-wavelet method and its application to obtain a reasonable,small number of elastic modes.
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