Effective Theories for Brittle Materials : a Derivation of Cleavage Laws and Linearized Griffith Energies from Atomistic and Continuum Nonlinear Models.
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Full text (MCPHS users only) |
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| Format: | Electronic eBook |
| Language: | English |
| Published: |
Berlin :
Logos Verlag Berlin,
2015
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| Series: | Augsburger Schriften Zur Mathematik, Physik und Informatik Ser.
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| Subjects: | |
| Local Note: | ProQuest Ebook Central |
MARC
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| 100 | 1 | |a Friedrich, Manuel. | |
| 245 | 1 | 0 | |a Effective Theories for Brittle Materials : |b a Derivation of Cleavage Laws and Linearized Griffith Energies from Atomistic and Continuum Nonlinear Models. |
| 260 | |a Berlin : |b Logos Verlag Berlin, |c 2015. | ||
| 300 | |a 1 online resource (294 pages) | ||
| 336 | |a text |b txt |2 rdacontent | ||
| 337 | |a computer |b c |2 rdamedia | ||
| 338 | |a online resource |b cr |2 rdacarrier | ||
| 490 | 1 | |a Augsburger Schriften Zur Mathematik, Physik und Informatik Ser. ; |v v. 28 | |
| 500 | |a 9.4.2 Step 2: Deformations with a finite number of cracks | ||
| 505 | 0 | |a Intro; 1 The model and main results; 1.1 The discrete model; 1.2 Boundary values and scaling; 1.3 Limiting minimal energy and cleavage laws; 1.4 A specific model: The triangular lattice in two dimensions; 1.5 Limiting minimal configurations; 1.6 Limiting variational problem; 1.6.1 Convergence of the variational problems; 1.6.2 Analysis of a limiting variational problem; 1.6.3 An application: Fractured magnets in an external field; 2 Preliminaries; 2.1 Elementary properties of the cell energy; 2.2 Interpolation; 2.3 An estimate on geodesic distances; 2.4 Cell energy of the triangular lattice | |
| 505 | 8 | |a 3 Limiting minimal energy and cleavage laws3.1 Warm up: Proof for the triangular lattice; 3.2 Estimates on a mesoscopic cell; 3.2.1 Mesoscopic localization; 3.2.2 Estimates in the elastic regime; 3.2.3 Estimates in the intermediate regime; 3.2.4 Estimates in the fracture regime; 3.2.5 Estimates in a second intermediate regime; 3.3 Proof of the cleavage law; 3.4 Examples: mass-spring models; 3.4.1 Triangular lattices with NN interaction; 3.4.2 Square lattices with NN and NNN interaction; 3.4.3 Cubic lattices with NN and NNN interaction; 4 Limiting minimal energy configurations | |
| 505 | 8 | |a 4.1 Fine estimates on the limiting minimal energy4.2 Sharp estimates on the number of the broken triangles; 4.3 Convergence of almost minimizers; 4.3.1 The supercritical case; 4.3.2 The subcritical case; 4.3.3 Proof of the main limiting result; 5 The limiting variational problem; 5.1 Convergence of the variational problems; 5.1.1 The Gamma-lim inf-inequality; 5.1.2 Recovery sequences; 5.2 Analysis of the limiting variational problem; 6 The model and main results; 6.1 Rigidity estimates; 6.2 Compactness; 6.3 Gamma-convergence and application to cleavage laws; 6.4 Overview of the proof | |
| 505 | 8 | |a 6.4.1 Korn-Poincaré-type inequality6.4.2 SBD-rigidity; 6.4.3 Compactness and Gamma-convergence; 7 Preliminaries; 7.1 Geometric rigidity and Korn: Dependence on the set shape; 7.2 A trace theorem in SBV2; 8 A Korn-Poincaré-type inequality; 8.1 Preparations; 8.2 Modification of sets; 8.3 Neighborhoods of boundary components; 8.3.1 Rectangular neighborhood; 8.3.2 Dodecagonal neighborhood; 8.4 Proof of the Korn-Poincaré-inequality; 8.4.1 Conditions for boundary components and trace estimate; 8.4.2 Modification algorithm; 8.4.3 Proof of the main theorem; 8.5 Trace estimates for boundary components | |
| 505 | 8 | |a 8.5.1 Preliminary estimates8.5.2 Step 1: Small boundary components; 8.5.3 Step 2: Subset with small projection of components; 8.5.4 Step 3: Neighborhood with small projection of components; 8.5.5 Step 4: General case; 9 Quantitative SBD-rigidity; 9.1 Preparations; 9.2 A local rigidity estimate; 9.2.1 Estimates for the derivatives; 9.2.2 Estimates in terms of the H1-norm; 9.2.3 Local rigidity for an extended function; 9.3 Modification of the deformation; 9.4 SBD-rigidity up to small sets; 9.4.1 Step 1: Deformations with least crack length | |
| 520 | 8 | |a Annotation |b A thorough understanding of crack formation in brittle materials is of great interest in both experimental sciences and theoretical studies. Such materials show an elastic response to very small displacements and develop cracks already at moderately large strains. Typically there is no plastic regime in between the restorable elastic deformations and complete failure due to fracture. The main focus of this book lies on the derivation of effective models for brittle materials in the simultaneous passage from discrete-to-continuum and nonlinear to linearized systems. In the first part the cleavage behavior of brittle crystals is investigated including the identification of critical loads for failure and the analysis of the geometry of crack paths that occur in the fractured regime. In the second part Griffith functionals in the realm of linearized elasticity are derived from nonlinear and frame indifferent energies by means of a quantitative geometric rigidity result for special functions of bounded deformation. | |
| 588 | 0 | |a Print version record. | |
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| 650 | 0 | |a Mathematical models. | |
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| 776 | 0 | 8 | |i Print version: |a Friedrich, Manuel. |t Effective Theories for Brittle Materials : A Derivation of Cleavage Laws and Linearized Griffith Energies from Atomistic and Continuum Nonlinear Models. |d Berlin : Logos Verlag Berlin, ©2015 |z 9783832540289 |
| 830 | 0 | |a Augsburger Schriften Zur Mathematik, Physik und Informatik Ser. | |
| 852 | |b E-Collections |h ProQuest | ||
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