Preface.- List of Symbols.- PART I Foundations of Continuum Damage Mechanics.- 1. Material Damage and Continuum Damage Mechanics.- 1.1 Damage and its Microscopic Mechanisms.- 1.2 Representative Volume Element and Continuum Damage Mechanics.- 2. Mechanical Representation of Damage and Damage Variables.- 2.1 Mechanical Modeling of Damage.- 2.2 Mechanical Representation of Three-Dimensional Damage State.- 2.3 Effective Stress and Hypothesis of Mechanical Equivalence.- 2.4 Elastic Constitutive Equation and Elastic Modulus Tensor of Damaged Material.- 2.5 Procedure of Continuum Damage Mechanics and its Refinement.- 3. Thermodynamics of Damaged Material.- 3.1 Thermodynamics of Continuum.- 3.2 Thermodynamic Constitutive Theory of Inelasticity with Internal Variables.- 3.3 Extension of Thermodynamic Constitutive Theory of Inelasticity.- 4. Inelastic Constitutive Equations and Evolution Equations of Material with Isotropic Damage.- 4.1 One-Dimensional Inelastic Constitutive Equation of Material with Isotropic Damage.- 4.2 Three-Dimensional Inelastic Constitutive Equations of Material with Isotropic Damage.- 4.3 Strain Energy Release Rate and Stress Criterion for Damage Development in Elastic-Plastic Damage.- 4.4 Inelastic Damage Theory based on Hypothesis of Total Energy Equivalence.- 5. Inelastic Constitutive Equation and Damage Evolution Equation of Material with Anisotropic Damage.- 5.1 Elastic-Plastic Anisotropic Damage Theory based on Second-Order Symmetric Damage Tensor.- 5.2 Elastic-Plastic Anisotropic Damage Theory in Stress Space.- 5.3 Fourth-Order Symmetric Damage Tensor and its Application to Elastic-Plastic-Brittle Damage.- PART II Application of Continuum Damage Mechanics.- 6. Elastic-Plastic Damage.- 6.1 Constitutive and Evolution Equations of Elastic-Plastic Damage - Ductile Damage, Brittle Damage and Quasi-Brittle Damage.- 6.2 Ductile Damage and Ductile Fracture.- 6.3 Application to Metal Forming Process.- 6.4 Analysis of Sheet Forming Limit by Anisotropic Damage Theory.- 6.5 Constitutive Equations of Void-Containing Ductile Material.- 6.6 Continuum Damage Mechanics Theory with Plastic Compressibility.- 7. Fatigue Damage.- 7.1 High Cycle Fatigue .- 7.2 Low Cycle Fatigue.- 7.3 Uncoupled Numerical Analysis of Very Low Cycle Fatigue.- 8. Creep Damage and Creep-Fatigue Damage.- 8.1 Creep Damage and Phenomenological Theory of Creep Damage.- 8.2 Viscoplastic Damage Theory of Creep Damage.- 8.3 Creep-Fatigue Damage.- 8.4 Effect of Damage Field on Stress Field at a Creep Crack Tip.- 9. Elastic-Brittle Damage.- 9.1 Damage of Elastic-Brittle Material.- 9.2 Isotropic Damage Theory of Concrete.- 9.3 Anisotropic Brittle Damage Theory by Second-Order Damage Tensor.- 9.4 Anisotropic Brittle Damage Theory with Elastic Modulus Tensor as Damage Variable.- 9.5 Anisotropic Brittle Damage Theory with Compliance Tensor as Damage Variable.- 10. Continuum Damage Mechanics of Composite Material.- 10.1 Damage of Laminate Composites.- 10.2 Elastic-Brittle Damage of Ceramic Matrix Composites.- 10.3 Local Theory of Metal Matrix Composites.- 11. Local Approach to Damage and Fracture Analysis.- 11.1 Local Approach to Fracture Based on Continuum Damage Mechanics and Finite Element Method.- 11.2 Mesh-Sensitivity in Time-Independent Deformation.- 11.3 Regularization of Strain and Damage Localization in Time-Independent Materials.- 11.4 Mesh-Sensitivity in Time-Dependent Deformation.- 11.5 Causes of Mesh-Sensitivity in Time-Dependent Deformation.- APPENDIX Foundations of Tensor Analysis.- A.1 Vectors and Tensors.- A.2 Vector Product, Tensor Product and the Components of Tensors.- A.3 Orthogonal Transformation, Invariants and Eigenvalues of Tensors.- A.4 Differentiation and Integral of Tensor Fields.- A.5 Differential Calculus of Tensor Functions.- A.6 Representation Theorem for Tensor Functions.- A.7 Matrix Representation of Tensors and Tensor Relations.- Reference Books and Bibliography.- Subject Index.

Continuum damage mechanics : a continuum mechanics approach to the analysis of damage and fracture

The mixture stress gradient theory of elasticity is conceived via consistent unification of the classical elasticity theory and the stress gradient theory within a stationary variational framework. The boundary-value problem associated with a functionally graded nano-bar is rigorously formulated. The constitutive law of the axial force field is determined and equipped with proper non-standard boundary conditions. Evidences of well-posedness of the mixture stress gradient problems, defined on finite structural domains, are demonstrated by analytical analysis of the axial displacement field of structural schemes of practical interest in nano-mechanics. An effective meshless numerical approach is, moreover, introduced based on the proposed stationary variational principle while employing autonomous series solution of the kinematic and kinetic field variables. Suitable mathematical forms of the coordinate functions are set forth in terms of the modified Chebyshev polynomials, satisfying the required classical and non-standard boundary conditions. An excellent agreement between the numerical results of the axial displacement field of the functionally graded nano-bar and the analytical solution counterpart is confirmed on the entire span of the nano-sized bar, in terms of the mixture parameter and the stress gradient characteristic parameter. The effectiveness of the established meshless numerical approach, demonstrating a fast convergence rate and an admissible convergence region, is hence ensured. The established mixture stress gradient theory can effectively characterize the peculiar size-dependent response of functionally graded structural elements of advanced ultra-small systems. 

On the analytical and meshless numerical approaches to mixture stress gradient functionally graded nano-bar in tension

The mixture stress gradient theory of elasticity is conceived via consistent unification of the classical elasticity theory and the stress gradient theory within a stationary variational framework. The boundary-value problem associated with a functionally graded nano-bar is rigorously formulated. The constitutive law of the axial force field is determined and equipped with proper non-standard boundary conditions. Evidences of well-posedness of the mixture stress gradient problems, defined on finite structural domains, are demonstrated by analytical analysis of the axial displacement field of structural schemes of practical interest in nano-mechanics. An effective meshless numerical approach is, moreover, introduced based on the proposed stationary variational principle while employing autonomous series solution of the kinematic and kinetic field variables. Suitable mathematical forms of the coordinate functions are set forth in terms of the modified Chebyshev polynomials, satisfying the required classical and non-standard boundary conditions. An excellent agreement between the numerical results of the axial displacement field of the functionally graded nano-bar and the analytical solution counterpart is confirmed on the entire span of the nano-sized bar, in terms of the mixture parameter and the stress gradient characteristic parameter. The effectiveness of the established meshless numerical approach, demonstrating a fast convergence rate and an admissible convergence region, is hence ensured. The established mixture stress gradient theory can effectively characterize the peculiar size-dependent response of functionally graded structural elements of advanced ultra-small systems.

Geometrically accurate structural analysis models in BIM-centered software

In this paper, soil-structure interaction problems are analysed using a three-dimensional formulation obtained by combining the Boundary Element Method (BEM) and the Finite Element Method (FEM). Structural elements that interact with the soil are modelled with the FEM. Soil, which is considered as a semi-infinite, homogeneous, linear elastic and isotropic medium, is modelled with the BEM using Mindlin's fundamental solution. A pile is discretized by several three-dimensional finite beam elements. A plate is discretized by finite flat shell elements, making it possible to consider the bending and membrane effects. The vertical and horizontal loads in the plate-pile-soil system are directly applied to the plate. This approach allows for a static analysis of the full interaction between capped pile groups or piled rafts and the soil. The developed formulations can also be extended to dynamic or non-linear analyses. Coupling between the BEM and FEM is performed through a mixed formulation in which the matrix of the soil's influence coefficients obtained by BEM is added to the stiffness matrix of the structural elements obtained by FEM, resulting in an equivalent stiffness matrix. Numerical examples of the plate-pile-soil interactions are solved, and their results are compared with those of other formulations.

A 3D BEM/FEM formulation for the static analysis of piled rafts and capped pile groups subjected to vertical and horizontal loads

In this paper, a practical boundary element method (BEM) is developed to analyze piled rafts. The raft is modeled using the direct boundary element formulation for thick plates; which is suitable for such type of problems. An innovative methodology to add stiffness matrices to the plate integral equations is developed. Hence, pile–soil–raft interactions are added as a special case of the developed methodology. In order to avoid generating huge systems of coupled equations, an effective condensation technique of piles and soil degrees of freedom (DOFs) at pile–soil–raft interface is employed. Several numerical examples are presented and results are compared to previously published results. Then, a practical application is solved to demonstrate the strength and versatility of the proposed formulation.

Practical boundary element method for piled rafts

Modeling nonlinearities, including damage, in the boundary element method (BEM) is usually carried out in implicit way, or in other words via applying initial stresses or strains over a discretized domain part. Such initial values have no physical meaning. They are only used to compensate the stress level due to the occurred nonlinearity. In this paper explicit implementation of nonlocal damage is proposed. The damaged points inside the domain is physically weakened by decreasing their modulus of elasticity. With the help of Eshelby's equivalent inclusion theory, this idea is developed and implemented in this work. Load control solution algorithm is used. Both average strain and average damage nonlocal models are considered. Numerical examples are presented to verify the developed formulation. Factors that affect the solution accuracy are studied in details.

Explicit boundary element modeling of nonlocal damage with Eshelby theory

This paper presents efficient analysis of plates on nonlinear foundations. The Reissner plate theory is used to model plates. Foundations are presented as the Winkler springs or the elastic half space. The developed analysis is mainly presented for tensionless foundation; however as demonstrated, it is straightforward extended to analysis of elastic-plastic foundations. The plate is analyzed using the boundary element method (BEM). Unlike the traditional BEM which uses equations in form ([H] {u} = [G] {t}), the presented formulation uses finite element like equations, in the form of ([K] {u} = {P}). An innovative formulation is presented to derive the relevant plate stiffness matrix [K] and load vector {P} from the BEM integral equation. Iterative procedures together with condensation process are used to eliminate degree of freedom at failed zones. Results of the present analysis are more accurate than those obtained from previously published results. The main advantages of the presented technique are its simplicity and accuracy and it gains both advantages of the boundary element and the finite element methods.

Efficient analysis of plates on nonlinear foundations

In this paper, new fundamental solutions for dynamic analysis of plane elastodynamics are developed. The governing dynamic differential equations are rewritten by replacing the accelerations with the corresponding displacements at different time steps via a suitable finite difference scheme. The known displacements in initial conditions or from previous time steps are treated as generalized new inertia terms. The unknown displacements at the current time step are added to the governing differential operator to form a new operator. The new time-dependent fundamental solutions are then derived with respect to the new differential operator. The required mathematical derivations are presented in detail. The corresponding integral equations and domain load treatments are also presented. Singular integrals of the derived fundamental solutions are also treated. Several numerical examples are solved to demonstrate the validity and the accuracy of the developed new solutions. The results demonstrated the accuracy of the developed new solutions compared to traditional time domain fundamental solutions. 

Time-Differencing Fundamental Solutions for Plane Elastodynamics

In this paper, alternative static-like fundamental solutions for transient dynamic analysis of three-dimensional elasticity are developed. The new formulation is based on modification of Navier's equations of elastodynamics by expressing the acceleration in terms of the displacements at different time steps using suitable finite difference scheme. The new fundamental solutions are derived via operator decoupling technique. The displacements from the previous steps are used to form the new generalized inertia terms. The new fundamental solutions are used to solve non-homogeneous media by dividing the continuous media into multi regions. The stiffness and load vector of each region are calculated using the boundary integral equation approach. Then, the whole multi-region system is solved as super finite elements. Numerical examples are solved to demonstrate the validation of the proposed solutions. The results of the proposed solutions are more stable and accurate than those of previous time domain and dual reciprocity formulations. 

Ahmed Fady Farid

Papers

Practical boundary element method for piled rafts

Explicit boundary element modeling of nonlocal damage with Eshelby theory

Efficient analysis of plates on nonlinear foundations

Time-Differencing Fundamental Solutions for Plane Elastodynamics

New three-dimensional time-stepping transient fundamental solutions with applications