Mechanics Today

Contributors

Preface

Contents of Volume

Summary

I Theory of Creep and Shrinkage in Concrete Structures: A Précis of Recent Developments

1. Introduction

2. Basic Experimental Facts Relating to Creep and Shrinkage

3. Concrete Approximated as an Aging Viscoelastic Material

3.1 Integral-Type Creep Law

3.2 Creep Function in Contemporary Recommendations by Engineering Societies

3.3 Dirichlet Series Expansions of Creep and Relaxation Functions

3.4 Rate-Type Creep Law

4. Environmental Factors and Nonlinear Effects in Creep and Shrinkage

4.1 Effect of Temperature and Humidity on Aging

4.2 Creep Law of Mass Concrete at Variable Temperature

4.3 Microdiffusion Mechanism of Creep and Its Thermodynamics

4.4 Constitutive Equation at Variable Temperature and Humidity

4.5 Nonlinear Effects in Creep and Shrinkage

4.6 Drying and Wetting of Concrete

5. Methods of Structural Analysis

5.1 Elastic-Viscoelastic Analogy for Aging Materials

5.2 Homogeneous Structures and McHenry's Analogy

5.3 Numerical Step-by-Step Methods

5.4 Conversion of Inelastic Strains to Applied Loads

5.5 Approximate Solutions Based on Simplified Creep Laws

6. Practical Problems in Design and Conclusion

7. References

8. List of Basic Notations

II On Nonequilibrium Thermodynamics of Continua

1. Introduction

2. Classical Concepts

2.1 First Law

2.2 State Space

2.3 Second Law

2.4 References

3. Nonlinear Continuum Mechanics

3.1 Conservation Laws

3.2 Constitutive Relations

3.3 Response Functionals

3.4 State Space and Hidden Coordinates

3.5 Internal Energy as a State Variable

3.6 Other Principles

3.7 References

4. Nonequilibrium Temperature and Entropy; the Second Law

4.1 Temperature

4.2 Entropy

4.3 Some Definitions

4.4 On Nonuniqueness of Nonequilibrium Entropy and Temperature

4.5 On the Principle of Nondecreasing Entropy

4.6 Bounds on Entropy

4.7 Heat Conduction

4.8 References

5. Thermodynamics of Materials with Memory

5.1 Representation of Memory by Means of Hidden Coordinates

5.2 On Path Parameter in State Space: A Theory of Viscoplasticity

5.3 Representation of Memory by Means of Response Functionals

5.4 On the Chain Rule of Differentiation

5.5 Representation in Terms of Response Functionals, and Clausius Inequality

5.6 Equilibrium Entropy

5.7 Nonequilibrium Entropy

5.8 References

6. Conclusions

7. References

III Mathematical Aspects of Finite-Element Approximations in Continuum Mechanics

1. Introduction

2. Weak Solutions of Elliptic Boundary-Value Problems

2.1 Introductory Ideas

2.2 Energy Spaces

2.3 Sobolev Spaces and Sobolev Imbeddings

2.4 Existence and Uniqueness

2.5 Variational Formulations

3. Approximations of Weak Solutions

3.1 Projections and Prolongations

3.2 Galerkin Approximations

3.3 Weighted Residuals

3.4 Ritz Approximations

3.5 Least-Square Approximations

3.6 Functional Imbeddings

4. Energy Convergence and Accuracy of Ritz-Galerkin Approximations of Elliptic Equations

4.1 Galerkin Approximations

4.2 Ritz Approximations

4.3 Nitsche's Trick

4.4 Consistency and Stability of Galerkin Approximations

5. Finite-Element Approximations

5.1 Introductory Comment

5.2 Connectivity of Discrete Models

5.3 Global and Local Representations

5.4 Finite-Element Models of Boundary-Value Problems

5.5 Some Distributional Considerations

6. Spline and Finite-Element Interpolation Spaces

6.1 Introductory Comment

6.2 L-Spline Spaces and One-dimensional Finite Elements

6.3 Rectangular Finite-Element Interpolation in E2

6.4 Prismatic, Cubic, and n-Dimensional Finite Elements

6.5 Triangular Finite Elements

6.6 Generalizations, Isoparametric Elements

7. Perturbation Errors

7.1 Quadrature Errors

7.2 Other Perturbation Errors, Conditioning and Boundary Errors

8. Nonlinear Problems

8.1 Introductory Comment

8.2 Nonlinear Elliptic Problems—Variational Approach

8.3 Strongly Monotone Operators

9. Mixed Finite-Element Approximations

9.1 Mixed Models

9.2 T*T Operators, Canonical Theory

9.3 Projections of the Canonical and Dual Equations

9.4 Consistency, Stability, and Convergence of Mixed Models

10. Initial-Value Problems

10.1 Introductory Comment

10.2 Linear Parabolic Problems

10.3 Some Fully Discrete Approximations

10.4 Nonlinear Second-Order Hyperbolic Equations

11. Closing Comments

12. References

IV Nonlinear Geometrical Acoustics

1. Introduction

2. Acceleration Fronts

3. Linear Geometrical Acoustics

4. Nonlinear Geometrical Acoustics

4.1 Weak Shocks

5. Modulated Simple Waves

6. Noninteracting Simple Waves

7. The Evolution of a Self-sustained Oscillation

8. A Resonant-Forced Oscillation

9. Nonlinear Interacting Waves

10. References

Author Index

Subject Index