Nonlinear Mechanics for Composite Heterogeneous Structures

Cover
Half Title
Title Page
Copyright Page
Dedication
Contents
Foreword
Preface
Acknowledgements
List of Figures
List of Tables
1. Introduction
	1.1. Introduction to composite materials
	1.2. Principles of continuum mechanics
		1.2.1. Elements from linear algebra: Vectors and matrices
			1.2.1.1. Vectors
			1.2.1.2. Matrices
		1.2.2. Tensors
		1.2.3. Additional operations and notations for tensors
		1.2.4. Deformation
			1.2.4.1. Lagrangian and Eulerian description of motion
			1.2.4.2. Deformation gradient tensor
			1.2.4.3. Green strain tensor
			1.2.4.4. Euler strain tensor
			1.2.4.5. Principal strains
		1.2.5. Stress
			1.2.5.1. Cauchy stress tensor
			1.2.5.2. First Piola-Kirchhoff stress tensor
			1.2.5.3. Second Piola-Kirchhoff stress tensor
			1.2.5.4. Principal stresses
			1.2.5.5. Stress invariants and deviatoric stress
		1.2.6. Constitutive description of the material behaviour
			1.2.6.1. Generalized Hooke’s law and elasticity tensor
			1.2.6.2. Material symmetry
			1.2.6.3. Isotropic materials
2. Linear and non-linear finite element analysis
	2.1. Introduction
	2.2. Equilibrium
		2.2.1. Strong form
		2.2.2. Energy principles and weak form
			2.2.2.1. Principle of virtual work
			2.2.2.2. Principle of stationary potential energy
	2.3. The finite element method
		2.3.1. Discretization
		2.3.2. Equilibrium within the finite element method
		2.3.3. Boundary conditions
		2.3.4. Isoparametric formulation
		2.3.5. Quadrilateral isoparametric element
		2.3.6. Numerical integration
	2.4. Non-linear finite element analysis
		2.4.1. The general framework of non-linear analysis
		2.4.2. Newton-Raphson incremental-iterative process
		2.4.3. Newton-Raphson under displacement control
		2.4.4. Geometrical non-linearity
			2.4.4.1. Continuum mechanics formulations
			2.4.4.2. Linearization of the governing equation
			2.4.4.3. Discretization of the governing equation
			2.4.4.4. Formulations for a quadrilateral element
		2.4.5. Material non-linearity
3. Failure of heterogeneous materials using non-linear continuum laws
	3.1. Introduction
	3.2. Plasticity
	3.3. Continuum damage mechanics
		3.3.1. Isotropic damage law
		3.3.2. Implementation of damage mechanics within finite element analysis
		3.3.3. Localization
		3.3.4. Enhancement of continuum damage mechanics to depict localization
		3.3.5. Cohesive zone model
		3.3.6. Implementation of cohesive zone models within finite element analysis
		3.3.7. Non-local damage model
4. Contact mechanics
	4.1. Introduction
	4.2. Theoretical framework: NSO, NSA, VI, HI and CP
		4.2.1. Smooth and non-smooth functions
			4.2.1.1. One-sided differentials
			4.2.1.2. Convex analysis
		4.2.2. Linear and non-linear complementarity problems
	4.3. Interaction along boundaries and interfaces
		4.3.1. Tie contact conditions
		4.3.2. Unilateral contact
		4.3.3. Frictional stick-slip effects
		4.3.4. Adhesive contact
	4.4. Boundary value problems in non-smooth mechanics
		4.4.1. Variational Inequalities
		4.4.2. Implicit variational inequalities
		4.4.3. The case of unilateral contact with friction
		4.4.4. Signorini-Coulomb unilateral frictional contact
		4.4.5. Computational algorithms
	4.5. Discretized Problems
		4.5.1. Optimization and contact analysis
		4.5.2. Existence of solution
5. The Extended Finite Element Method
	5.1. Introduction to linear fracture mechanics
		5.1.1. The stress intensity factor
		5.1.2. Energy balance and the energy release rate
		5.1.3. J-integral and the interaction integral
	5.2. The Extended Finite Element Method (XFEM) within linear fracture mechanics
		5.2.1. Enrichment for weak discontinuities
		5.2.2. Enrichment for strong discontinuities
		5.2.3. Discretization of the governing equations within linear fracture mechanics
		5.2.4. Crack propagation criteria
		5.2.5. Crack tip enrichment approaches and numerical integration
	5.3. The XFEM method for cohesive cracks
		5.3.1. Definition and weak formulation
		5.3.2. Discretization and linearization of the governing equation
		5.3.3. Crack propagation and numerical integration for cohesive cracks
6. Homogenization
	6.1. Introduction to the concept of homogenization
	6.2. Asymptotic expansion homogenization
		6.2.1. Microscale description
		6.2.2. Macroscale description
		6.2.3. Implementation within finite element analysis
		6.2.4. A brief on mathematical homogenization for non-linear problems
	6.3. Numerical homogenization
7. Multi-scale analysis for composite materials
	7.1. Introduction to multi-scale computational homogenization
	7.2. Formulation of first-order computational homogenization
		7.2.1. Averaging relations
		7.2.2. The overall multi-scale formulation
			7.2.2.1. Prescribed displacements
			7.2.2.2. Periodic boundary conditions
	7.3. An introduction to multi-scale schemes adopted to simulate localization of damage
	7.4. A multi-scale model proposed to simulate localization of damage in composite structures
8. Data-driven analysis
	8.1. Introduction to data-driven numerical simulation
	8.2. Data-driven analysis using direct interpolation from databases
	8.3. Data-driven analysis incorporating machine learning
		8.3.1. Soft computing, artificial intelligence and neural networks
			8.3.1.1. Feed-forward neural networks
			8.3.1.2. Deep learning
			8.3.1.3. Available tools
		8.3.2. Machine learning applications in engineering mechanics
			8.3.2.1. Constitutive material modelling
			8.3.2.2. Consideration of thermodynamic restrictions
			8.3.2.3. Multi-scale modelling and scale-bridging
			8.3.2.4. Transfer learning
			8.3.2.5. Postprocessing and zooming
			8.3.2.6. Physics-informed neural networks
			8.3.2.7. Integration of machine learning tools in multi-scale FE2-like procedures
			8.3.2.8. Emerging fields
	8.4. Data-driven analysis based on the minimum distance among points of the data set
Appendix A: Matlab codes on numerical modelling of composite heterogeneous structures
	A.1. Multi-scale computational homogenization (FE2)
	A.2. Data-driven computational homogenization (FE2) using the concept of minimization of a distance function
	A.3. Data-driven computational homogenization (FE2) using machine learning principles
	A.4. A multi-scale scheme proposed to simulate localization of damage in composite structures using the XFEM method and principles of contact mechanics
Bibliography
Index