Differential Geometry Approach to Continuous Model of Micro-Structural Defects in Finite Elasto-Plasticity
Abstract
:1. Introduction
2. Metric Tensor and Deformation Gradient
- 1.
- the set of n-dimensional differentiable manifolds with ;
- 2.
- the set of diffeomorphisms of the class called the motion of the body,
2.1. Metric and Induced Inner Product
2.2. Deformation Gradient and Its Adjoint
- 1.
- Let us remark that (which means that the components of the Cauchy–Green tensor are the metric components) if and only ifThus, the metric components of the differential manifolds are related by the components of the deformation gradient, considered
- 2.
- By definition of the Cauchy–Green tensor, (27), while the metric tensor, denoted here by applied to a vector field is a dual vector, namely as it follows from (8). Thus, only if is identified with can we say that a metric tensor can be viewed as a Cauchy–Green tensor. Moreover, from (30) the metric tensor with the components induces the metric with the components by the deformation gradient, represented in terms of its components
- 3.
- 4.
- In classical continuum mechanics, all vector fields refer to the same vector space, which is associated with the Euclidean space The dual space is identified with the vector space. Consequently, the adjoint and the transpose of a tensor coincide. Moreover, the metric and non-metric properties are independent feature of the connection.
- 5.
- All the formulas derived above remain valid if the deformation gradient is replaced by an anholonomic diffeomorphism.
3. Material Connection-Revisited Decomposition Theorem
3.1. Covariant Derivatives of the Tensor Fields
3.2. Decomposition Theorem of the Metric Connection
- The connection is metric;
- There exists an anholonomic diffeomorphism such that the metric tensor
- a.
- a connection compatible with the anholonomic diffeomorphism i.e., in the gradient notation
- b.
- such that is a second-order symmetric tensor
- c.
- such that i.e., is 2-form such that
- 1.
- The contortion and the torsion are tensors of the type , which determine each other bytaking into account the definition of the torsion
- 2.
- The following skew-symmetries hold:
4. Plastic Connection for Material with Micro-Structure
- -
- and are differentiable manifolds, at any time t;
- -
- the existence of a differentiable manifold (time dependent, but we omit the t in the description for the sake of simplicity), called intermediate configuration, such that and are diffeomorphic; i.e., there exists a diffeomorphism, say of the class for all and at every time
- -
- the deformation gradient is induced by the motion function, i.e.,
- -
- the existence of the plastic and elastic non-induced diffeomorphisms, and (called distortions),
- The connection is metric;
- There exists a diffeomorphism such that the metric tensor is given by ;
- The metric is induced from by via the Formula (66).The following representation for the plastic connection coefficients is provided as
- a.
- is a connection compatible with the diffeomorphism i.e., in the gradient notation ;
- b.
- such that is a second-order symmetric tensor
- c.
- such that i.e., is two-form
5. Measure of Microstructural Defects
5.1. Density of Dislocations
5.2. Density of Disclinations
5.3. Lattice Defects and Their Interplay
- (i)
- The point defect characterized byIn this case ; thus, no contribution to the torsion follows.
- (ii)
- The gradient-type defect occurs if there exists a symmetric, second-order field, such that
6. Conclusions
Funding
Conflicts of Interest
References
- Mandel, J. Plasticité Classique et Viscoplasticité; CISM-Udine; Springer: Vienna, Austria; New York, NY, USA, 1972. [Google Scholar]
- Teodosiu, C. A dynamic theory of dislocations and its applications to the theory of the elastic- plastic continuum. In Fundamental Aspects of Dislocation Theory; Simmons, J.A., De Witt, R., Bullough, R., Eds.; National Bureau of Standards (U.S.): Gaithersburg, MD, USA, 1970; Volume 317, pp. 837–876. [Google Scholar]
- Cleja-Ţigoiu, S.; Soós, E. Elastoviscoplastic models with relaxed configurations and internal state variables. Appl. Mech. Rev. 1990, 43, 131–151. [Google Scholar] [CrossRef]
- Romanov, A.E. Mechanics and physics of disclinations in solids. Europ. J. Mech. A/Solids 2003, 22, 727–741. [Google Scholar] [CrossRef]
- Cleja-Ţigoiu, S.; Paşcan, R.; Ţigoiu, V. Interplay between continuous dislocations and disclinations in elasto-plasticity. Int. J. Plast. 2016, 68, 88–110. [Google Scholar] [CrossRef]
- Cleja-Ţigoiu, S.; Paşcan, R.; Ţigoiu, V. Disclination based model of grain boundary in crystalline materials with microstructural defects. Int. J. Plast. 2019, 114, 227–251. [Google Scholar] [CrossRef]
- De Wit, R. Theory of Disclinations: II. Continuous and Discrete Disclinations in Anisotropic Elasticity. J. Res. Nat. Bur. Stand. A Phys. Chem. 1973, 77A, 49–100. [Google Scholar] [CrossRef] [PubMed]
- Fressengeas, C.; Taupin, V.; Capolungo, L. An elasto-plastic theory of dislocation and disclination field. Int. J. Solids Struct. 2011, 48, 3499–3509. [Google Scholar] [CrossRef] [Green Version]
- Fressengeas, C.; Taupin, V.; Capolungo, L. Continuous modeling of structure of symmetric tilt boundaries. Int. J. Solids Struct. 2014, 51, 1434–1441. [Google Scholar] [CrossRef]
- De Wit, R. A view of the relation between the continuum theory of lattice defects and non-Euclidean geometry in the linear approximation. Int. J. Eng. Sci. 1981, 19, 1475–1506. [Google Scholar] [CrossRef]
- Kröner, E. The Differential geometry of Elementary Point and Line Defects in Bravais Crystals. Int. J. Theor. Phys. 1990, 29, 1219–1237. [Google Scholar] [CrossRef]
- Cleja-Ţigoiu, S. Anisotropic Damage in Elasto-plastic Materials with Structural Defects. In Multiscale Modelling in Sheet Metal Forming; Banabic, D., Ed.; Spinger International Publishing AG: Cham, Switzerland, 2016; pp. 301–351. [Google Scholar]
- Cleja-Ţigoiu, S.; Ţigoiu, V. Continuous Model of Structural Defects in Finite Elasto-Plasticity; Éditions Universitaires Européenes: Paris, France, 2017; pp. 5–73. [Google Scholar]
- Cleja-Ţigoiu, S. Finite Elasto-Plastic Models for Lattice Defects in Crystalline Materials. In Mathematical Modelling in Solid Mechanics, Advanced Structured Materials 69; dell’Isola, F., Sofonea, M., Steigmann, D., Eds.; Springer Nature Singapore Pte Ltd.: Singapore, 2017; pp. 43–58. [Google Scholar]
- Cleja-Ţigoiu, S. Evolution Equation for Defects in Finite Elasto-Plasticity. In Generalized Models and Non-classical Approaches in Complex Materials 1, Advanced Structured Materials 89; Altenbach, H., Pouget, J., Rousseau, M., Collet, B., Michelitsch, T., Eds.; Springer International Publishing AG.: Cham, Switzerland, 2018; pp. 179–201. [Google Scholar]
- Gurtin, M.E.; Fried, E.; Anand, L. The Mechanics and Thermodynamics of Continua; Cambridge University Press: Cambridge, MA, USA, 2010. [Google Scholar]
- Cleja-Ţigoiu, S. Elasto-plastic materials with lattice defects modelled by second order deformations with non-zero curvature. Int. J. Fract. 2010, 166, 61–75. [Google Scholar] [CrossRef]
- Nakahara, N. Geometry, Topology and Physics, 2nd ed.; Taylor & Franci: New York, NY, USA; London, UK, 2003. [Google Scholar]
- Lee, J.M. Introduction to Smooth Manifolds; Springer Science+Business Media: New York, NY, USA, 2013. [Google Scholar]
- Noll, W. Materially Uniform Simple Bodies with Inhomogeneities. Arch. Rat. Mech. Anal. 1967, 27, 1–32. [Google Scholar] [CrossRef]
- Wang, C.C. On the Geometric Structure of Simple Bodies, a Mathematical Foundation for the Theory of Continuous Distributions of Dislocations. Arch. Rat. Mech. Anal. 1967, 27, 33–94. [Google Scholar] [CrossRef]
- Steinmann, P. Geometrical Foundations of Continuum Mechanics, An application to First- and Second- Order Elasticity and Elasto-Plasticity; Springer: Berlin/Heidelberg, Gremany, 2015. [Google Scholar]
- Epstein, M. The Geometry Language of Continuum Mechanics; Cambridge University Press: Cambridge, MA, USA, 2010. [Google Scholar]
- Clayton, J.D. Defects in Nonlinear Elastic Crystals: Differential Geometry, Finite Kinematics, and Second-Order Analytical Solutions. ZAMM—J. Appl. Math. Mech./Z. Angew. Math. Und Mech. 2015, 95, 476–510. [Google Scholar] [CrossRef]
- Yavari, A.; Goriely, A. Riemann-Cartan Geometry of Nonlinear Disclination Mechanics. Math. Mech. Solids 2012, 18, 91–102. [Google Scholar] [CrossRef]
- Clayton, J.D. Nonlinear Mechanics of Crystals; Solid Mechanics and Its Applications 177; Springer: Berlin/Heidelberg, Germany, 2011. [Google Scholar]
- Yavari, A.; Goriely, A. Riemann-Cartan Geometry of Nonlinear Dislocation Mechanics. Arch. Rat. Mech. Anal. 2012, 205, 59–118. [Google Scholar] [CrossRef]
- Cartan, H. Calcul Différentiel. Formes Différentielles; Hermann: Paris, France, 1967. [Google Scholar]
- Bilby, B.A. Continuous distribution of dislocations. In Progress in Solid Mechanics; Sneddon, I.N., Hill, R., Eds.; North-Holland Publishing Company: Amsterdam, The Netherlands, 1960; pp. 329–398. [Google Scholar]
- Minagawa, S. A non-Riemannian geometrical theory of imperfections in a Cosserat continuum. Arch. Mech. 1979, 31, 783–792. [Google Scholar]
- Le, E.; Stumpf, H. On the determination of crystal reference in nonlinear continuum theory of dislocations. Proc. R. Soc. Lond. A 1996, 452, 359–371. [Google Scholar]
- Kröner, E. The internal mechanical state of solids with defects. Int. J. Solids Struct. 1992, 29, 1849–1857. [Google Scholar] [CrossRef]
- Kleman, M.; Fridel, J. Disclinations, dislocations, and continuous defects: A reappraisal. Rev. Mod. Phys. 2008, 80, 61–115. [Google Scholar] [CrossRef] [Green Version]
- Cleja-Ţigoiu, S. Disclinations and GND tensor effects on the multislip flow rule in crystal plasticity. Math. Mech. Solids 2020, 25, 1643–1676. [Google Scholar] [CrossRef]
- Schouten, J. Ricci Calculus; Springer: Berlin, Germany, 1954. [Google Scholar]
- Cleja-Ţigoiu, S. Material Forces in Finite Elastoplasticity with Continuously Distributed Dislocations. Int. J. Fract. 2007, 147, 67–81. [Google Scholar] [CrossRef]
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Cleja-Ţigoiu, S. Differential Geometry Approach to Continuous Model of Micro-Structural Defects in Finite Elasto-Plasticity. Symmetry 2021, 13, 2340. https://doi.org/10.3390/sym13122340
Cleja-Ţigoiu S. Differential Geometry Approach to Continuous Model of Micro-Structural Defects in Finite Elasto-Plasticity. Symmetry. 2021; 13(12):2340. https://doi.org/10.3390/sym13122340
Chicago/Turabian StyleCleja-Ţigoiu, Sanda. 2021. "Differential Geometry Approach to Continuous Model of Micro-Structural Defects in Finite Elasto-Plasticity" Symmetry 13, no. 12: 2340. https://doi.org/10.3390/sym13122340
APA StyleCleja-Ţigoiu, S. (2021). Differential Geometry Approach to Continuous Model of Micro-Structural Defects in Finite Elasto-Plasticity. Symmetry, 13(12), 2340. https://doi.org/10.3390/sym13122340