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Mechanical Constitutive Models for Engineering Materials

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Mechanical Constitutive Models for Engineering Materials MECHANICAL CONSTITUTIVE MODELS FOR ENGINEERING MATERIALS by Behzad Rohani Soils and Pavements Laboratory U. S. Army Engineer Waterways Experiment Station P. O. Box 631, Vicksburg, Miss. 39180 September 1977 Final Report Approved For Public Release; Distribution Unlimited TA Prepared for Assistant Secretary of the Army (R&-D) 7 Department of the Army ,W34m Washington, D. C. 203I0 S-77-19 1977 Under Project 4 A I6 II0 IA 9 ID LIBRARY JAN 2 1970 Bureau of Reclamation Pnnvo*' f*- »^ ¡-~ J Destroy this report when no longer needed. Do not return it to the originator. V BUREAU OF RECLAMATION DENVER LIBRARY 92051290 V Unclassified ß ‘[ SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) READ INSTRUCTIONS REPORT DOCUMENTATION PAGE BEFORE COMPLETING FORM 1 .^REPORT NUMBER 2. GOVT ACCESSION NO. 3. RECIPIENT'S CATALOG NUMBER vb Miscellaneous^Paper S-77-19 A 4. T IT L E (and Subtitle) ' 5. TYPE OF REPORT & PERIOD COVERED 3 MECHANICAL CONSTITUTIVE MODELS FOR ENGINEERING £ Final Report) MATERIALS'-— .--------------------- ------ $ 6. PERFORMING ORG. REPORT NUMBER 7. A U T H O R S 8. CONTRACT OR GRANT NUMBER^) ( Behzad Rohani /" 4 9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT, PROJECT, TASK AREA & WORK UNIT NUMBERS U. S.^Army Engineer Waterways Experiment Station Soils and Pavements Laboratory ^ ^ Project UA161101A91B P. 0. Box 631, Vicksburg, Miss. 39180 11. CONTROLLING OFFICE NAME AND ADDRESS / 12. REPORT DATE Assistant Secretary of the Army (R&D) ’ September 1977 t/ Department of the Army, Washington, D. C. 20310 13. NUMBER OF PAGES ' l8l 14. MONITORING AGENCY NAME & ADDRESSf/f different from Controlling Office) 15. SECURITY CLASS, (of this report) Unclassified 15a. DECLASSI FI CATION/DOWNGRADING SCHEDULE 16. DISTRIBUTION STATEM ENT (oi this Report) Approved for public release; distribution unlimited. 17. DISTRIBUTION STATEM ENT (of the abstract entered in Block 20, if different from Report) 18. SUPPLEMENTARY NOTES 19. KEY WORDS (Continue on reverse side if necessary and identify by block number) Constitutive equations Constitutive models Elastic media Plastic media Viscoelastic media 20. ABSTRACT ÇContfnue en reverse aixte ft neceasary and identify by block number) Some of the basic mathematical tools and physical concepts necessary for development of mechanical constitutive relationships are reviewed and presented in orthogonal Cartesian coordinate system, using both indicial and matrix notations. Following this review, various classes of isotropic constitutive relationships are discussed and examined. The analyses are restricted to deformation for which displacement gradients are small. The (Continued) DD »ET» 1473 EDITION OF I MOV 65 IS OBSOLETE Unclassified SECURITY CLASSIFICATION OF THIS PAGE (’Rîrert Data Entered) ___________ Unclassified___________ SECURITY CLASSIFICATION OF THIS PAGEfWhan Data Entered) 20. ABSTRACT (Continued). constitutive relationships are classified and presented in the following categories: _a. Constitutive equations of elastic materials, h. Incremental constitutive equations. c_. Constitutive equations of simple viscoelastic materials. _d. Constitutive equations of plastic materials. Unclassified SECURITY CLASSIFICATION OF THIS PAGEfWhen Data Entered) PREFACE This investigation was conducted "by the U. S. Army Engineer Waterways Experiment Station (WES) under Department of the Army Project UAI6IIOLA.9ID, ^In-House Laboratory Independent Research Program, sponsored hy the Assistant Secretary of the Army (R&D). The investigation was conducted by Dr. B. Rohani during the calendar years 1975 and 1976 under the general direction of Messrs. J. P. Sale, Chief, Soils and Pavements Laboratory, and Dr. J. G. Jackson, Jr., Chief, Soil Dynamics Division. The report was written by Dr. Rohani. Directors of WES during the investigation and the preparation of this report were COL G. H. Hilt, CE, and COL J. L. Cannon, CE. Tech­ nical Director was Mr. F. R. Brown. 1 CONTENTS Page PREFACE . ................................................. 1 PART I: INTRODUCTION.......................................... 3 Background ...................................... 3 Objective . ............................................... 3 S c o p e ........................... k PART II: MATHEMATICAL PRELIMINARIES ........................... 5 Indicial-Notation ........................................ 5 Matrix Algebra ........................................... 10 Solutions of Linear Algebraic Equations .................. 17 Eigenvalue Problem ...................................... 18 Cayley-Hamilton Theorem .................................. 2k Cartesian Tensors .................... 26 PART III: SUMMARY OF BASIC CONCEPTS FROM CONTINUUM MECHANICS . 1+5 Stress Tensor.................. 1+5 Strain Tensor ............................................ 53 Strain-Rate Tensor . ..................................... 6l Equations of Continuity and M o t i o n ...................... 6k Constitutive Equations .................................. 65 PART IV: CONSTITUTIVE EQUATIONS OF ELASTIC MATERIALS .......... 67 Cauchy’s Method ........................................... 67 Green’s Method............ 93 PART V: INCREMENTAL CONSTITUTIVE EQUATIONS .................... 109 Incremental Constitutive Equation of Grade-Zero ........... 112 Incremental Constitutive Equation of Grade-One ........... 113 Incremental Constitutive Equation of Grade-Two ........... Il6 Variable-Moduli Constitutive Models ................... 118 PART VI: CONSTITUTIVE EQUATIONS OF SIMPLE VISCOELASTIC MATERIALS............................................ 136 Kelvin-Voigt Material .................................. 137 Maxwell Material........................................ ll+0 Standard-Linear Material ................................ lk2 PART VII: CONSTITUTIVE EQUATIONS OF PLASTICITY . ............ 1I+7 Ideal Plastic Material .................................. ikQ Work-Hardening Plastic Material ........................... 16k REFERENCES ..................... 173 APPENDIX A: SELECTED BIBLIOGRAPHY ............................ A1 TABLE A1 2 MECHANICAL CONSTITUTIVE MODELS FOR ENGINEERING MATERIALS PART I : INTRODUCTION Background 1. Development of mechanical constitutive models (defined as load-deformation or stress-strain relationships) for engineering ma­ terials has received considerable attention in recent years, particu­ larly in the field of geotechnical engineering. The primary reason for such efforts is the fact that with the advent of high-speed electronic computers and the development of new methods of numerical analysis, a variety of complex engineering problems can be solved provided realistic constitutive relationships for the materials of interest are available. Stress-strain relationships for a number of materials, such as soil, rock, and concrete, are often nonlinear even when the magnitudes of the strains involved are small. This type of nonlinear behavior, referred to as physical nonlinearity, has been the subject of investiga­ tion at the U. S. Army Engineer Waterways Experiment Station (WES) since early i9 60 ; special emphasis has been placed on modeling the mechanical behavior of earth materials. During the fall of 19715 an elementary course on mechanical constitutive relationships was offered at the Vicksburg Graduate Center, WES, and a series of lecture notes was pre­ pared for use by the students taking this course. The purpose of the lecture notes was to acquaint the students with some of the basic physical concepts and mathematical tools available for developing con­ stitutive relationships. The lecture notes were purposely kept to an elementary level, and were prepared with the formulation of constitutive relations for earth materials in mind. Objective 2. The objective of this report is to document the lecture notes 3 in a format that can "be used for engineering training throughout the Corps of Engineers, U. S. Army, or as materials for self-study and reference pruposes. Scope 3. Some of the basic mathematical tools necessary for the develop­ ment of constitutive relationships are presented in Part II. Included in Part II are: a brief discussion of indicial notation, matrix algebra, development of basic equations related to eigenvalue problem, the Cayley-Hamilton theorem, and Cartesian tensors (with emphasis on second-order tensors). A number of numerical examples are included in this part of the report in order to help the reader to better understand the subject matter. Part III includes a summary of appropriate equa­ tions from continuum mechanics required for this elementary presenta­ tion of the subject of constitutive relationships. Constitutive equations of elastic materials are developed in Part IV. The so- called incremental constitutive equations are discussed in Part V. Constitutive equations of simple viscoelastic materials are discussed in Part VI. Constitutive equations of plasticity are contained in Part VII. 1* PART II: MATHEMATICAL PRELIMINARIES Some of the “basic mathematical tools necessary for treatment and understanding of the physical concepts to he presented in the en­ suing parts of this report are developed in this part. The development is kept to an elementary level and is confined to orthogonal Cartesian coordinate system. In order to establish a common basis of terminology and notation, both indicial and matrix notations are briefly discussed. However, indicial notation is used for most of the presentations throughout this report in order to keep the number of equations to a minimum. Indicial Notation 5. The
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