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
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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)
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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)
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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 development of indicial notation is based on a number of agreements motivated by miniaturization of a large system of equations or variables. For example, if three variables are denoted by X1 , 5 and X^ , we can simply denote them by X_^ , where the subscript i is called an index and we agree that it takes on values 1, 2, and 3 (three-dimensional geometry). Similarly, the system of equations A-^ = X^ + , A^ = X^ + Y.^ , and A^ = X^ + Y^ can be expressed as A^ = X^ + Y^ . An index which is not repeated in any single term is called a free index. Thus, the index i in X. and A. = X. + Y. is ------1111 a free index. Furthermore, a free index must appear in every term of an expression. Systems which depend on one free index, such as Xi and A^ , are called systems of first order. The terms X^ , X^ , and X^ are called the components or elements of the system. A first- order system, therefore, has three components. Systems which depend on two free indices, such as A^ , are called systems of second order. Since the indices take on values 1, 2, and 3* a second-order system has nine components. Similarly, we can define systems of third order which depend on three free indices and have twenty-seven components, e.g., A^ • this report, however, we will be dealing mainly with first- and second-order systems.
5 6. If an index appears twice in a term it is called a dummy index. For example, the index i in is a dummy index. By agreement, a dummy index implies that the term is to he summed with respect to this index over the range of the index. Thus Aii A11 + A22 + A33 ’ XiYi = X1Y1 + X2Y2 + X3Y3 ’ and Sij = ° - Ei3 = A B C (1) m r mr B = D E (2) r mr m In the first expression the index m is free and the index r is a dummy. In the second expression the index r is free and the index m is a dummy. The index r in Equations 1 and 2 is called a connecting index. If we substitute the second expression into the first expres sion and use the same letters for indices, we obtain A = D E C (3) m mr m mr Equation 3 is meaningless since it is not consistent with the rules (agreements) of indicial notation; the index m appears three times on the right-hand side of this expression. To obtain the correct ex pression we must first overhaul the dummy index m in Equation 2. Using the useful characteristic that the particular letter used for a dummy index is immaterial, we can write B = D E (b) r pr p 6 Substituting Equation k into Equation 1 we obtain A = D E C (5) m pr p mr Equation 5 is notationally correct; there is no question as to which index is the free index. Expanding the dummy indices p and r over their range, Equation 5 takes the following form A = D J C . + D J C „ + D J C , m pi p ml p2 p m2 p3 p m3 = D.-E-C n + DnoEnC _ + DnoEnC , 11 1 ml 12 1 m2 13 1 m3 + D21E2Cml + D22E2Cm2 + E23E2Cm3 + D31E3Cml + D32E3Cm2 + E33E3Cm3 (6 ) Equation 6 (or Equation 5) has three components. The first component, for example, becomes A1 = Dl A Cll + D12E1C12 + D13E1C13 + D21E 2C11 + D22E2C12 + D23E2C13 (T) + D31E 3C11 * E 32E3C12 + D33E 3C13 which is quite long in comparison with the compacted indicial form. 7. Another agreement in establishing indicial notation is the use of commas in the subscripts to represent partial derivatives. Thus, we agree that 8F F . (8a) 3X. l ,1 3U. (8b) 9X7 = u i,j J 7 Similarly dU 8U •p = __ 2L = TT ri (8c) nk dX n dX1 k m,n s ?m,k In Equation 8c, m is a dummy index and n and k are free indices. Expanding the dummy index m , Equation 8c takes the following form E = U, IL . + U0 U0 . + U0 U_ nk l,n l,k 2,n 2,k 3,n 3,k (9) Equation 9 (or Equation 8c) has nine components. For example, the E 13 component becomes 10 E13 U1,1U1,3 + U2,1U2,3 + U3,1U3,3 ( ) 8. In indicial notation the condition of symmetry of a second- order system is denoted by B. . = B.. (1 1 ) ij Ji The condition of skew-symmetry is denoted by C.. = -C.. (12) ij Ji Equation 11 results in conditions B12 B21 conditions Eh = B_^ (13) 23 32 of symmetry B„ = B 31 13 8 whereas Equation 12 indicates that C11 = C22 = C33 = ° C12 = -C21 conditions of skew- symmetry (lU) C23 C32 = -C '31 13 Using the above conditions, an' asymmetric (i.e., neither symmetric nor skew-symmetric) second-order system can be expressed as the sum of a symmetrical system 1/2(T. . + T..\ and a skew-symmetrical system / V \ J1/ l/2[T.. - T..\ , i.e. , T. . = 1/2/t . . + T. A (15) iJ V 1J J 1/ 9. In using indicial notations, one often deals with quantities that have no free index. Such quantities are referred to as scalars or zero-order systems. For example, the following quantities are scalars A \ k A B mn nm (16) D D D mn np pm It is noted that all indices in Equation 16 are dummy indices. The ex panded form of the last expression in Equation 1 6 , for example, becomes 9 D D D = D D D,+D0D D 0 +D_D D _ mn np pm In np pi 2n np p2 3n np p3 = DniDn D .. + Dn0 D0 D n + D 1 0 D0 D . 11 lp pi 12 2p pi 13 3p pi + D0 D-. D 0 + D0 0 D0 D 0 + D0 JD0 D _ 21 lp p2 22 2p p2 23 3p p2 + D D D + D D D + D _D D 0 31 lp p3 32 2p p3 33 3p p3 DllDlir)ll + D11D12I)21 + D11D13D31 + D12D21D11 + D12D22D21 + D12D23D31 + D13D31I311 + D13D32r)21 + D13D33I)31 + D21D11D12 + D21D12D22 + D21D13D32 + D22D21D12 + D22D22I)22 + D22D23D32 + D23I)31I)12 + D23D 32D22 + D23D 33I)32 + D31D11EI13 + D 31'D12I)23 + D 31D13'D33 + r)32I)21D13 + D32D22D23 + D32D23D33 (17) + D33D31D13 + D33D32D23 + D33D33D33 The compactness of indicial notation is once again demonstrated by the above expansion. Matrix Algebra 10. Another convenient method for representing a large number of equations or quantities is through matrix notation. A matrix is an array of numbers or components of a system. For example, the components of a first-order system can be arranged as 10 [X] = [xx X2 X3] (18b) Equation l8a represents a 3-by-l (3 rows and 1 column) column matrix whereas Equation l8b represents a l-by-3 (l row and 3 columns) row matrix. Similarly, the components of a second-order system can "be arranged as A A ^ A ^ 11 12 13 [A] A A„ (19) 21 22 23 A A ^ A„ 31 32 33 Equation 19 represents a 3-hy-3 square matrix. We are mainly interested in 3-by-3 matrices in this report. Some useful types of matrices are: a. Diagonal matrix in which all elements other _ than those on the diagonal are zero. 0 0 11 0 0 (20a) [B] 22 0 0 B 33 h. Unit matrix in which all off-diagonal elements are zero and every diagonal term is unity. 100 H J = 0 10 (201d ) 0 0 1 c_. Symmetrical matrix in which off-diagonal terms are symmetrical. B11 B12 B13 [B] = B B^ B„ (20c) 12 22 23 B B^„ B__ 13 23 33 jd. Skew-symmetrical matrix in which every diagonal term is zero and off-diagonal terms are skew- symmetric. 0 C12 C13 [C] = 0 (20d) -C12 C23 o i 1 CVJ on o 0 H 00 In indieial notation the counterparts of Equations 20c and 20d are given by Equations 11 and 12, respectively. Similarly, in indicial form Equation 20a can be expressed as B . . = 0 for i ^ j . * 1J 11. The transpose [A] of a square matrix [A] is obtained by completely interchanging every row with its corresponding column: AAA 11 21 31 [A]* = AAA 12 22 32 (21) AAA 13 23 33 12 * In indicial notation A. = A . . In view of Equations 11 and 12, in 1J Jl ^ the case of a symmetric matrix B.. = B . . , and in the case of a skew- * XJ x0 symmetric matrix C.. = -C.. . ij ij 12. Matrices obey certain prescribed rules of matrix axgebra. Addition or subtraction of matrices having the same number of rows and the same number of columns is accomplished by adding or subtracting corresponding elements. For example, consider the following 3-by-3 matrices: _&31 a32 a33 b 11 12 13 b ^ b b 21 22 23 (22b) Td _ b b 31 32 33 Two 3-by-3 matrices can be obtained by adding or subtracting matrices [a] and [b] ; thus, all + bll a12 + b12 a13 + b13 [c] = [a] + [b] a21 + b21 a22 + b22 a23 + b 23 (23a) a31 + b31 a32 + b32 a33 + b33 1 U* H all H a12 b12 al3 b13 [d] = [a] - [b] - h22 - b23 a21 “ b21 a22 a23 (23b) - h32 - b33 a31 - b31 a32 a33 13 In indicial notation the second-order systems [c] and [d] can he ex pressed by c.. = a.. + b.. and d..=a..-b... A matrix can be * ij ij ij iJ multiplied by a number k by simply multiplying every element in the matrix by k . Two matrices can be multiplied together if they are comformable, i.e., if the number of columns of the first matrix is equal to the number of rows of the second. A p-by-q matrix and a q-by-s matrix are conformable and can be multiplied together. The result of multiplication is a p-by-s matrix. For example, consider the multipli cation of matrices [a] and [b] given in Equation 22: [a][b] = [e] (2U) The matrix [e] is a 3-by-3 matrix whose components are obtained from the following rule, expressed in indicial notation, governing matrix multiplication: (25) For further examples of matrix multiplication consider the following: [a]2 = [m] [a]3 = [n] [a][b][a] = [p] (26) [a]2 [b] = [q] [a]2 [b]2 = [r] [a][b][a]*[b] = [s] lb ! Using the rule governing matrix multiplication (Equation 25) it follows that the components of matrices [m] , [n] , [p] , [q] , [r] , and [s] take the following forms m. . ij n. . ij = aikakfafj = a., b. „an . Pij lk kf fj (27) = a., a, „b„. lk kf fj r. . = a... a., „b„ b . ij lk kf fg gj * s. . = a .. b., „a „ b . ij lk kf fg gj * Note that in the last expression in Equation 27 the definition a^ = a ^ is invoked. Furthermore, it should he noted that in Equations 25 and 27 the indices i and j are the only free indices. 13. The sum of the diagonal terms of a square matrix is called the trace of the matrix and is denoted by tr (e.g., trace of [a] = tr[a] = a ^ + a22 + a ^ ) . In indicial form, tr[a] = a. . (28) li Similarly, in view of Equations 25 through 27, tr([a][b]) a., b . lk ki tr[a]^ • n . £L lk ki (2 9 ) tr[a]^ aikakfafi tr([a][b][a]) aikbkfafi 15 All indices in Equation 29 are dummy indices, indicating that the trace of a matrix is a scalar. Also, tr([a][b]) = tr([b][a]) even though [a][b] # [b][a] . This can be verified by expanding the indicial form aik"^ki ^ i k ^ i ih. The determinant of a square matrix is denoted by det[a] , or simply |a| , and is expressed as (for a 3-by-3 square matrix) H H a12 a13 CVJ on a21 a22 all^a22a33 " a23a32^ a12^a2la33 a23a31^ (30) a31 a32 33 + a13(a2ia32 " a22a31^ It is noted that the determinant of a matrix is also a scalar. In con junction with the determinant of a matrix we define the minor and cofactor. The minor of an element a . . of the matrix [a] is obtained th th ^ by deleting the i row and j column and forming the determinant of the remaining terms. For example, the minor of a^g element is given as a21 a23 minor of a, (31) a21a33 a23a31 a31 a33 The cofactor of an element a . . is the minor of that element with a sign attached to it according to the following criterion cofactor of a . . = (-l)1+^ minor of a . . (32) ij ij Thus, the cofactor of a ^ element is given as 1+2 cofactor of a±2 = (-1) (a^a^ - a23a31) -(a2ia33 - a23a31) (33) In view of the definition of minor and cofactor the determinant of 16 the matrix [a] can he expressed as | a | = a1 1 ( cofactor of a ^ ) + a]_2^coi>ac^or of a.^) + a^ ( cofactor of a ) (3U) It should he pointed out that Equation 3^ is not unique in calculating the determinant of the matrix [a] . The same final products will re sult from expansion on columns or other rows, e.g., |a| = a^Ccofactor of a ^ ) + a22^coi*ac^or a22^ + a^^(cofactor of a ^ ) (35) 15. Finally, we define the inverse [a] ^ of a square matrix [a] such that [a]-1[a] = [a][a]-1 = flj■ (36) The inverse matrix is given hy [a]'1 = (3T) where the matrix [A] , called the adjoint of [a] , is determined by * replacing the elements of [a] hy their corresponding cofactors; thus, 22 33 a32a23 a32a13 ” a12a33 a12a23 " a22a13_ [A] = (38) a31a23 " a21a33 alla33 " a31a13 a21a13 " alla23 _a21a32 ” a31a22 a31a12 " alla32 alla22 " a12a21_ From Equation 37 it follows that the inverse exists provided a| ^ 0 . Solutions of Linear Algebraic Equations 16. Consider a set of linear algebraic equations 17 + + II allxl a12x2 a13X3 H II + + M C a21xl a22x2 a23X3 (39) II + + CO a31xl a32x2 a33X3 In indicial notation Equation 39 can be expressed as a..x. = k. (UO) ij J i In matrix notation Equation 39 takes the following form [a]{x} = {k} (bl) where [a] is a square matrix of coefficients, {x} is a column matrix of unknowns, and {k} is a column matrix with known elements. The objective is to solve for the elements of the column matrix {x} . Pre multiplying both sides of Equation 1)1 by [a]-^ results in [a] 1 [a]{x} = [a] ^{k} (1)2 ) or, in view of Equation 36, t'I'Hx} = {x} = [a]-1{k} (1)3) From Equation 1)3 it follows that once the inverse of the coefficient matrix is determined the solution for {x} is obtained by performing the indicated matrix multiplication. Eigenvalue Problem IT. In a number of engineering problems the following system of algebraic equations is often encountered ([a] - XflJHx} = {0} (tt) 18 ■where the elements of the column matrix {x} are the unknowns to he determined, A is a scalar parameter, and {0} is a null column matrix (all elements being zero). According to Equations H3 and 37, a non trivial solution of Equation k h exists only if the determinant of the coefficient matrix vanishes, i.e., £L_ _ all - x 12 a13 = 0 a21 ^22. """ ^ a23 (U5) a31 32 a33 " X The expansion of the above determinant yields the following cubic equa tion in X : X 3 - I 2 + II À - III = 0 a x a a (U6) where I tr[a] a a nn (V ra) II = sum of the minors of the diagonal elements of [a] a a22 a23 all a13 all a12 + + (UTb) a32 a33 a31 a33 a21 a22 III =|a| (Vfc) a 1 1 Equation k6 is called the characteristic equation of the matrix [a] . 18. The three roots A_^ , A^ , and A^ of the characteristic equation are called the characteristic values or eigenvalues of [a] . For every eigenvalue A^ (assuming that all three roots are distinct) Equation H6 is satisfied and hence Equation HU has nontrivial solutions: 19 xli ixU^)} = ^ x2i (U8) X3i, Every such solution of {x} is called a characteristic vector or eigenvector of [a] . The eigenvectors {x(X^)} corresponding to eigenvalues X^ -can he grouped together to form a square matrix re ferred to as a modal column matrix, i.e., X11 X12 X13 [x] = X21 x22 x23 (U9) X31 X32 x33 For each eigenvalue and the corresponding eigenvector, Equation' h k can be written as [a]{x(Xi)} = A^{x(X_ )} (50) Since [a] is a 3-by-3 matrix, Equation 50 can he expressed in the following form for all of the eigenvalues X , , and X^ • | 1 O all al2 al3 X11 X12 x13 O X11 x12 x13 H --- = a21 a22 a23 X21 X22 x23 X21 x22 x23 0 x2 0 (51) 0 0 X3 a31 a32 a33 X31 X32 x33 X31 X32 x33 - “ - _ Equation 50 is also satisfied if each eigenvector is multiplied hy an arbitrary constant c^ , i.e., [a]c^{x(X^)} = c^X^ixiX^)} (52) Therefore, an eigenvector is indeterminate to the extent that it can be multiplied by an arbitrary constant. Selecting an eigenvector 20 x. ' l i li { y ( A . )} = (53) i 2i = Ci < X 2i X 3i 3i appropriate to eigenvalue , the corresponding modal column matrix becomes CO 1 n 1—1 H H yl 2 C1X11 C2X12 C3X13 = [y] = y21 y22 y23 Clx21 C2X22 C3X 23 _y31 y32 y 33_ .C1X 31 C2X32 C3X 33. 1 X CO 1 — 0 0 ‘ "Xll X12 1 'C1 = 0 c2 0 X21 X22 X23 0 (5k) .X31 X 32 X 33_ 0 °3. It is observed from Equation 5U that a modal column matrix is indeter minate to the extent that it can be postmultiplied by a diagonal matrix of arbitrary constants c^ . Now utilizing the modal column matrix [y] , Equation 51 can be expressed as [a][y] = [y]|XI (55) where [A] is a diagonal matrix with elements ^ , and . Premultiplying Equation 55 by [y]_1 we obtain [y]_1 [a][y] = [AJ (56) Postmultiplying Equation 55 hy [y]-1 gives [a] = [y ] ["Aj [y ]-1 (57) From Equation 56 it is observed that the modal column matrix [y] which is found by grouping the eigenvectors of [a] diagonalizes the 21 matrix [a] . Furthermore, the elements of the diagonalized matrix are the eigenvalues of [a] . This diagonalization process is an important part of the eigenvalue problem and its significance will he realized when dealing with second-order systems. 19. As an example of an eigenvalue problem, consider the follow ing system of equations (2 - X)x1 - x2 + x^ = 0 -8x1 + (3 - *)x2 + 7*3 = 0 -8x^ - x2 + (ll - l)x^ = 0 In matrix form the above system of equations is expressed as ’ 2 -1 1 "loo" -8 3 7 -X 0 1 0 _-8 -1 11_ _0 0 !_ or (see Equation bk) ([a] - XtlJ'Hx} = {0} The characteristic equation of [a] is given as (See Equations i+5 and U6) 2 - X -1 1 -8 3 - X 7 = -X3 + l6X2 - 68X + 80 = 0 -8 -1 11 -■ X ; noted that I = 16 II = 68 , and III = 80 . Solution a a ’ a of the characteristic equation yields the following eigenvalues for the matrix [a] : X1 = 2 , X2 = k , *3 = 10 22 For each value of A there exists three homogeneous equations. For A = A^ = 2 we have -8x^ + Xg + = 0 - 8 x 1 - x 2 + 9 X J = 0 where it is noticed that = 1 , xg = 1 , x3 = 1 is a nontrivial solution. The eigenvector corresponding to A^ then becomes (see Equation 53) = c -l \ 1 Similarly, for A = A = U , {y(A2 )> = c2 <-l and for A = A^ = 10 , = c3 \ ! The modal column matrix becomes (see Equation 5*0 23 ci c2 0 [y] = C1 -c2 C3 C1 c2 c3 Using Equation 56 it can be verified that the modal column matrix trans forms the matrix [a] into a diagonal matrix with elements A , A,. 2 5 and A^ , i.e. - 1 i C-J^ c 2 0 2 - 1 1 O o 2 0 0 ro H ______^t o O -8 3 T = C1 _C2 c3 C1 -°2 c3 O o o o d o ( ( ( ( -8 -1 11 r 0 0 10 1 * Cayley-Hamilton Theorem 20. The Cayley-Hamilton theorem plays an important role in ex pressing higher powers of square matrices. It simply states that a square matrix satisfies its own characteristic equation. The result of the theorem is given here without proof. Let [a] be a 3-by-3 matrix and its characteristic equation be given as (see Equation U6) X3 - I A2 + II X - III = 0 (58) a a a If [a] satisfies its characteristic equation it follows that [a]3 = III [ U - II [a] + I [a]2 (59) Note that the constant III is multiplied by a unit matrix [l] . a From Equation 59 it follows that [a]U = [a]3[a] = I III [l] + (III - I IIn)[a] + (i2 - II_)[a]2 (60) O t £L c l cL gL y d ci/ / 2l+ Similarly, [a]5 = [a]1*!»] = - I H aXIa)tlO ♦ (laH I a - IaIIa + n a)tal + (Ta - 2 Ian a + IIIa) [a]2 <6l) It is clear from the examples given in Equations 60 and 6l that using the Cayley-Hamilton theorem ,(i.e., Equation 59) we can express any power of [a] greater than 3 in terms of [a] and [a] . Accordingly, a polynomial representation of [a] , i.e., [g] = f ([a]) = k0MJ + k^a] + k2[a]2 + k^a]3 +...+ k j a f (62) where kQ , k^ , ..., k^ are constants, can he expressed as [g] = + n1 [a] + n2 [a]2 (63.) where the coefficients tIq > 9 an<^ ^2 are now P°lynomial functions of I , II , and III . a a 5 a 21. For an illustrative example of the Cayley-Hamilton theorem, consider the following matrix 12 0 [c] 3 -1 -2 10-3 The characteristic equation of [c] is given as X3 + 3A2 - 7A - IT = 0 where it is noted that III = IT c 25 We substitute [c] for A in the characteristic equation and multiply the constant term by a unit matrix , i.e., 3 2 1 2 0 1 2 0 1 2 0 3 -1 -2 +3 3 -1 -2 -7 3 -1 -2 _ 1 0 - 3_ _ 1 0 - 3_ _ 1 0 - 3 . _ 1 0 0 3 ll+ 12 21 0 - 12 - ■17 0 1 0 = 27 -11 -38 + -6 21 2 k 0 0 1 13 -6 -31 -6 6 27 " 7 1 U o’ 17 0 o" 0 0 O" - 21 -7 - 11*. - 0 17 0 = 0 0 0 7 0 -21 0 0 17 0 0 0 - resulting in a null matrix [O] . Cartesian Tensors Cartesian coordinate 22. Let us consider the orthogonal Cartesian coordinate system xk (Figure l) with unit vectors ^ , i£ , and i3 along the x1 , x^ , and x^ axes, respectively. Figure 1. Orthogonal Cartesian coordinate system x^ 26 From elementary vector analysis the dot products of these unit vectors are given as (6U) or, in indicial notation, 0 p ^ r i i ( 65) r P 1 p = r This product is denoted by 6 ^ and is known as the Kronecker delta; thus, 0 P + r i • i = 6 (66 ) p r pr 1 p = r The counterpart of 6 in matrix notation is the unit matrix Did (see Equation 20b). From Equations 65 and 66 it follows that (67) 6 6 =3 pr rp Transformation matrix 23. The vector V with components (x^ , Xg , x^) in the x^ coordinate system (Figure 2) can be expressed in vector form as V x 1 ( 6 8 ) X111 + X212 + X313 P P If we fix the origin and rotate the axes forming a new coordinate system x^ , with corresponding unit vectors (Figure 2), then the vector 27 V with components (x^ , x£ , x p in the primed (rotated) system can "be denoted by V = xTiT s s (69) Figure 2. Orthogonal Cartesian coordinate systems and x^ In view of Equations 68 and 69 x i x'i (TO) P P s s The dot product of Equation TO with i results in K 28 Since i • = 6^k , Equation 71 reduces to x, = x !i! • in (7 2 ) K s s k By the definition of dot product, i 1 • i-. = COS (X * , X,) (73) s k s K where cos(x^ , x^.) is the cosine of the angle between the x^ and xfc axes. We denote cos(x^ , x ^ hy agk (7U) ask = oos(xs • *k> In view of Equations 73 and 7^-» Equation 72 takes the form. = a , x' (75) *k sk s Similarly, the dot product of Equation 70 with ik results in the following relation x a, x (76) k ks s Equation 75 relates the components of the primed system (rotated) to the components of the unprimed system. Equation 76 relates the com ponents of the unprimed system to the components of the primed system. The matrix a is called the transformation matrix and consists of sk the following table of direction cosines: Table of Direction Cosines X1 X2 X3 cd cd 1— 1 C\J 0 0 1 — 1 ! xi all ! x2 a21 22 a23 cd PO i — a_ ; x s 1 32 a33 where = cos(x^ , x^) , a^i = cos(x^ , x^) , a^2 = cos(x^ , x2), etc. 2k. Equations 75 and j6 can now he utilized to establish certain properties of the transformation matrix. Differentiating Equation 75 with respect to x^ yields ^ ,i askXs,i (77) Since x^ i = ôki , i.e., x1 1 = 1 , x1 2 = 0 , etc., Equation 7? reduces to (78) 6,ki . = a sk . x' s,i . From Equation 76, x' = a x , and thus sm m x f . = a x . = a 6 . (79) s, i sm m,i sm mi Substituting Equation 79 into Equation 78 results in 6. . = a _ a 6 . ( ) ki sk sm mi 80 In view of the definition of 6 ^ , Equation 80 reduces to a . a . = 6, . (81) sk si ki 30 Similarly, "by differentiating Equation 76 and folio-wing the same pro cedure we get a, a . = . (82) kp ip ki Equations 8l and 82 describe the basic properties of the transformation matrix. Expanding Equation 8l yields 2 2 2 = 1 all + a21 + a31 611 2 2,2 .. (83a) a12 + a22 a32 1 2 A 2 2 _ a13 a23 a33 1 = 0 all a12+ a21 a22 + a31a32 512 (83b) all al3 + a21 a23 + a31 a33 ° al2 al3 + a22 a23 + a32 a33 ° Similarly, expanding Equation 82 yields 2 2 2 all + a12 + a13 " 1 2 L 2 ^ 2 , (8Ua) *21 + a22 + a23 " 1 2 ^ 2 . 2 _ a31 a32 a33 1 CM l—1 + + 11 a12 a22 a13 a23 = 0 I a + a ^ a_ + a, „ = 0 (81+b) 11 31 12 32 13 33 o II a3 + + 0 0 CO '21 a31 a22 a32 a23 Equations 83a and 8ba indicate that the sum of the squares of the elements of any column or'now of the transformation matrix is unity and 31 called normalization conditions. Equations 83b and 8Ulo indicate that the sum of the products of corresponding elements in any two distinct columns or rows is zero and called orthogonality conditions. Through algebraic manipulations of Equations 83 and 8k it can also he shown that |a|2 = 1 (85) 25. For a numerical example of a transformation matrix consider the following rotation (Figure 3) of the x^ coordinate system: x0 X2 X / / / XL Figure 3. Rigid-body rotation of x^ coordinate system The transformation matrix a _ associated with this rotation can he sk constructed easily: £L Æ 0 2 2 /2 / 2 _ = cos(x^ , xfc) = sk ~ ° 0 0 1 32 It can readily "be verified that the above matrix satisfies Equations 83 through 85- First-order tensor 26. If with a coordinate transformation x,1 = a.u. x (see Equa- ks s tion 76), the three quantities in the unprimed coordinate system transform to three quantities in the primed reference frame by (86) ^ = vs A s then A is a tensor of the first order. We already know that A is s s a vector. Therefore, a vector is a tensor of first order. Within the context of indicial notation, a first-order tensor is a first-order system, i.e., it has one free index. Any quantity whose value does not v change with coordinate transformation is called a tensor of order zero or a scalar (see Equation l6). A scalar is, therefore, invariant to rigid-hody rotation of the coordinate system. Considering the scalar product of A£ with itself we obtain A'Al = a. A a. A (87) k k ks s kp p Since a, a = A£A£ = A A 6 (88 ) s p sp In view of the definition of 6 , Equation 88 reduces to sp (89) ^ SAAs s Equation 89 indicates that the sum of the square of the elements (com ponents) of a first-order tensor (vector) is invariant to rigid-body rotation of the coordinate axes. This quantity is the only invariant associated with a first-order tensor. The magnitude or length of the vector A is given as \j A A and is, therefore, invariant to rigid- body rotation of the coordinate system. 33 27. For an example of transformation of a first-order tensor, consider the vector with components in the coordinate system. The magnitude of the vector is = y j ( 5)2 + (IO)2 + (2 )2 = V 129" If the coordinate system undergoes a rigid-hody rotation as shown in Figure 3j the components of the vector in the rotated system can he calculated from Equation 86, i.e., A* = a., _ An + a_ + a., ^A^ 1 11 1 12 2 13 3 = f (5) + 4 (10) = 1 5 ^ A' = a_., A, + a „ A ^ + a „ A ^ 2 ^21 1 23 3 = (5) + ~ (10) = A' = a^.A., + a „ A ~ + a^^A^ 3 31 1 33 3 It is noted that the magnitude of the vector is not affected hy the co ordinate transformation, i.e., Second-order tensor 28. Consider two first-order tensors u. and v. associated 1 1 with coordinate system x. . Since u. and v. are first-order 1 11 tensors we may write (see Equation 75) u. = a .u' (90a) 1 ni n v. = a .v* 1 1 mi m (90b) 3 ^ Combining the vectors and v_^ we can construct the second-order system u.v. , which we may call the array T.. , i.e., J 1J V l uxv2 ulv3 u.v. = T. . = 1 J ij U2V1 U2V2 U2V3 (91) U 3V1 U 3V 2 U 3V 3 In view of Equation 90 the product u.v. can he written as J T. . = u.v. = a .u!a .v1 = a .a .ufv f (92) ij i j ni n mj m ni mj n m Equation 92 provides the array of nine-number T . Denoting the array u !v ! by TT , Equation 92 becomes n m nm T. . = a .a .T1 ij ni mj nm (93) where is referred to the primed coordinate system. Similarly, starting from u! = a. u and v! = a. v , we can derive l in n l ìm m T ! . = a. a. T ij m jm nm (9*0 Any quantity T that transforms according to Equation 9*+ is called a second-order tensor. Within the context of indicial notation, a second-order tensor is a second-order system, i.e., it has two free indices. Accordingly, the addition, subtraction, and multiplication of second-order tensors are governed by the rules expressed in Equa tions 23 through 27. In matrix notation the transformation laws (Equations 93 and 9*0 are expressed as [T] = [a]*[T'][a] (95) [T-] = [a][T][a] (96) 35 where [a] is the transpose of [a] . 29. The second-order tensor is an extremely important tensor in mechanics and will be used extensively in this report. In particular, we are interested in second-order symmetric tensors such as stress and strain tensors. It was shown in Equation 89 that there is one invariant associated with a first-order tensor (vector). In the case of a second- order symmetric tensor, however, there are three independent quantities that remain constant with respect to coordinate transformation. These independent invariants are IT = tr[T] = T.. (97) IIT = tr[T] =T.A . (98) III-,T = tr[T]J = T., lk T,km Tmi . (99) From Equation 9^ it follows that tr[T"] = T! . a. a. T (100) 11 in ìm run According to the property of the transformation matrix (Equation 8l), a. a. = 6 , and Equation 100 becomes m lm nm tr[T1] = 6 T nm nm (101) In view of the definition of 6 , Equation 101 reduces to nm tr[T!] = T = tr[T] (102) nn indicating that tr[T] is an invariant. Similarly, from Equation 9 b 9 T!, = a. a. T , T ’ . = a, a. T , and lk m km nm ki iq? is ps 5 36 tr[T’]2 - ml m ! ik ki = a. a. T a. a. T in km nm kp is ps = a. a. T a. an T (103) m is nm km kp ps Again using the property of the transformation matrix (Equation 8l), a a = 6 , a, a, = S , and Equation 103 becomes in is ns ’ km kp mp tr[T']2 = 6 T 6 T (lOU) ns nm mp ps In view of the definitions of 6ng and 6 , Equation 10U reduces to tr[T']2 = T T = tr[T]2 (105) sm ms p indicating that tr[T]- is an invariant. Using the same procedure it can "be shown that tr[T']3 = tr[T]3 (106) indicating that tr[T] is also an invariant. 30. The three invariants of the second-order tensor (i^ , II^ ? Ill ) can he related to the coefficients in the characteristic equation of the tensor (Equation UT)• By algebraic manipulation it can he shown that II (107a) T II (10Tb) T Now, using the Cayley-Hamilton theorem (Equation 59) in indicial form, i.e. , T T T III™«;. II T + I T. T . (108) ik km mj T 1J T ij T in nj 3T and taking the trace of the tensor (putting i = j), we obtain T. T T . = 3111^ - H mT.. + ImT. T . lk km mi T T n T m ni (109) In view of Equations 97 9 98, 99? and 107* Equation 109 results in IIIm = 3 H I t - 3IIt It + Ij (110a) 1 ------1 — 1 3 IIIm = — TTT _ — TT T 4 . _ TJ (110b) 3 T 2 T"T 6 T Equations 107 and 110 indicate that the coefficients in the character istic equation of the tensor are also invariant. 31. For an illustrative example of transformation of second-order tensors, consider the following tensor associated with an x co- K ordinate system: h 1 2 T. . = 1 6 o ij 2 0 8 From Equations 979 98, and 99 we have IT = tr[T] = 18 II = tr[T] = 126 IIIT = tr[ t ]3 = 966 Also, from Equation hf, IIT = 99 IIIT = l6o where it is noted that Equations 107 and 110 are satisfied. If the coordinate system undergoes a rigid-body rotation, such as the one 38 shown in Figure 3, the components of the tensor in the (rotated) system can he calculated from the transformation law of second-order tensors (Equation 9^)* The transformation matrix associated with the coordinate rotation in Figure 3 is given as 2 2 /2 /2 a sk 2 2 0 0 1 From Equation 9^ it follows that T! . a. a.nT + a. a.„T 0 + a. a.0T „ ij in jl nl m j2 n2 m j3 n3 a.na.,Tnn + a.0a.,T0.. + a.„a.-.Ton ll jl 11 i2 jl 21 i3 jl 31 + ailaj2T12 + ai2aj2T22 + ai3aj2T32 + a.na.0T,_ + a.0a.0T00 + a._a.0T00 ll j3 13 i2 j3 23 i3 j3 33 Substituting for the components of T. and a we obtain 1J SK 6 1 /2 l 1+ - / 2 /2 - / 2 8 Now, utilizing Equations 97» 98» 99» and we obtain II II I mtT t H 00 H 1 1 i i T ' = 126 = Ti t IIIT, = 966 = 39 arxta il rnfr T it a ignl om (.. T! . (i.e., . ! T form diagonal a into T. . transform will that matrix by Equation Equation by tensor. A diagonalization process was previously demonstrated for for demonstrated previously was process diagonalization A tensor. esr n h eeet o T!. r cle h picplvle o the of values principal the called are . ! T of elements the and tensor transformation a seek we tensors, second-order of law transformation the i r cle h rnia ae (r rnia drcin) f the of directions) principal (or axes principal the called are xi be normalized. The modal column matrix of normalized eigenvectors is is eigenvectors normalized of matrix column modal The normalized. be = 0 for i ^ j) associated with an x£ coordinate system. The axes axes The system. coordinate x£ an with associated j) ^ i for 0 = 3—by— 3 matrices in conjunction with the eigenvalue problem. It was was It problem. eigenvalue the with conjunction in matrices 3 3—by— length or magnitude of the eigenvector, then the eigenvector is said to to said is eigenvector the then eigenvector, the of magnitude or length h arx T cnb rte as written be can [T] matrix the lmns nte , oriae ytmae ie a T. . as given are system coordinate x, the in elements constant. If the arbitrary constant is chosen to be the inverse of the the of inverse the be to chosen is arbitrary constant an by arbitrary the If multiplied be can it that constant. extent the to determinate eigenvectors of a square matrix, diagonalizes the matrix as indicated indicated as matrix the the diagonalizes grouping by matrix, found is square a of which matrix, eigenvectors column modal the that shown law of a second-order tensor (Equation (Equation tensor second-order a of law second-order tensors. Consider a second-order symmetric tensor whose tensor symmetric second-order a Consider tensors. second-order quantities. these of nature invariant the indicating pose of the normalized modal column matrix is the transformation matrix transformation the is matrix column modal normalized the of pose column matrix by [y] , the diagonalization relation (Equation 56) for for 56) (Equation relation modal normalized the diagonalization , the [y] Denoting by matrix. matrix column column modal normalized a called In view of the properties of a transformation matrix, the transformation transformation the matrix, transformation a of properties the of view In oprsn fEuto 1 ihEuto 1 idcts ht h trans the that indicates 111 Equation with 112 Equation of Comparison K 32. We now proceed to establish some useful relationships for for relationships useful some establish to proceed now We 32. 56 Frhroe i a son ht n ievco i in is eigenvector an that shown was it Furthermore, . lJ J1 l J k [T* ] , T m A = [y]_ = DAJ H H J 1-3 10 III, = 160 = T H = 99 = ho 1 [T][u] 96 [T][a]* cnb rte as written be can ) T. . Using . .. T = ( (in) 112 ) which transforms T. . into a diagonal form. Furthermore, the elements of the diagonalized matrix are the eigenvalues of T ^ . In the case of second-order symmetric tensors, the eigenvalues (principal values) are always real. It should he noted that the normalization of eigenvectors is necessary in order to conform with the normalization conditions of the transformation matrix (Equations 83a and 8Ua). 33. For a numerical example of diagonalization of a second-order symmetric tensor consider the tensor T.. whose elements in the x, ij k coordinate system are given as -2 2 10 T. . 2 -11 8 ij 10 8 -5 The characteristic equation of is given as -2 - A 2 10 1 H 2 H 8 = (X - 9)(X + 9)(X + 18) = 0 10 8 -5 The eigenvalues of T^^ are, therefore, .*1=9; = -9 ; a3 = -18 Next, we determine the normalized eigenvectors for T^ . For A = A^ we can write down (see Equation kb) r—1 £ O II -llx + 2x 2 + CO 2x, - 20xo + 8x0 = 0 1 2 3 lOx + 8x 2 - 1 ^ 3 = 0 Solving the above system of equations and considering the normalization Ul 2 2 2 condition of the eigenvector (i.e., x^ + Xg + x^ = l), the normalized eigenvector corresponding to X becomes (y(x1 )} = Similarly, for A = A^ and A = A^ , we obtain {y(x2)> =< | iy(A3)} = The normalized modal column matrix then becomes 2 2 _ r ' 3 3 3 2. 2 3 "3 2_ 1 2 3 3 3 and h2 2 1 2 3 3 3 2_ 2 1 [yf = 3 3 3 1 £ 2 3 3 3 As was stated previously, the transformation matrix which transforms into a diagonal form is the transpose of the normalized modal column matrix. This can "be verified by using [y] as the transformation matrix [a] in-Equation 96, i.e., [T1] = [a][T][a]* = [y]*[T][v] 1 2 2 1 2 2 -2 2 10 3 3 3 3 3 3 2 2_ 1 1 2 2 2 -11 8 3 3 3 3 3 3 1 2 2 2 1 2 10 8 -5 3 3 3 3 3 3 Performing the above matrix operation we obtain 0 0 0 0 9 \ [T-] = 0 = 0 0 0 -9 X2 0 0 -18 0 0 X3_ 3k. Consider three second-order symmetric tensors , B 3 and C Using the Cayley-Hamilton theorem it was shown previously rs that a polynomial representation relating the components of two tensors takes the form given in Equation 63. In indicial notation Equation 63 is expressed in the following form. 1+3 A .. = f. .(B ) = ru<5. . + . (113) ij ij mn 0 ij n-B. 1 ij r)2BikBkj The counterpart of Equation 113 expressing the components of one tensor in terms of the components of two other tensors was derived by Rivlin and Ericksen.^ The Rivlin-Ericksen equation given here without proof has the following form A. . f. .(B , C ) = ruô. . + tuB. . + n0B.n B. . ij ij mn rs 0 ij 1 ij 2 lk kj + rue. . + n»,c.. c, . + nc(B., c. . + cM b. .) 3 ij b ik kj 5 ik kj lk kj + iv(B B C . + C..B, B .) 6 ik kp pj ik kp pj + n™(B., C. C . + C.. C. B . ) 7 ik kp pj ik kp pj' + r)o(B.. B C ,C, . + C..C. B B . ) o ik kp pt tj ik kp pt tj (llU) where the coefficients t]q , ... 5 rig are polynomial functions of the invariants of B and C^s and the following joint invariants mn n 1n = B ab ba n2 ^ab^bc^ca (115) ~ = B , K C n3 ab be ca i = B C n4 ab be cd da It is noted that when dependence on C disappears, Equation Ilk re duces to Equation 113. Equations 113 and llU are the bases for most of the presentations in the ensuing parts of this report. kk PART III: SUMMARY OF BASIC CONCEPTS FROM CONTINUUM MECHANICS Stress Tensor In Cartesian coordinate system x. ■we define the stress 35 . i tensor a. . at a point as ij F. a. . limit (116) ij A. 0 i where F. is force in the coordinate direction j and A is the area normal to i ^ axis on which the force F. acts. Figure U depicts the J Figure Stress components positive directions of the components of the stress tensor. In the absence of distributed body or surface couples the stress tensor is symmetrical, i.e., a.. = a.. . Accordingly, the state of stress at a point can be described by six independent stress components. Invariants of stress tensor 36. Stress tensor is a second-order symmetric tensor and it obeys the transformation law given in Equation 9 U, i.e., a! . = a. a. a ij m jm nm (117) where a! is referred to x! (rotated) coordinate system. As was 1 J 1 shown in Part II, a second-order tensor has three independent invariants (Equations 975 98, and 99)* In the case of stress tensor we define these invariants as I = a a nn (118) 2 aikaki (119) 1 J zr o..o. a . 3 3 lk km mi (120) Stress deviation tensor 37• Stress tensor can be expressed as the sum of two second- order symmetric tensors in the following manner o . . = S . . + 1 o <5. . ij ij 3 nn ij (121) where the tensor S.. = a.. - \ a 6. . (122) ij ij 3 nn ij is referred to as the stress deviation tensor and a 6 /? is nn ij called the spherical stress tensor. An important property of the stress deviation tensor is that its trace is equal to zero, i.e., S.. = a.. - a = 0 li li nn (123) The stress deviation tensor, therefore, has only two independent invari ants. We denote these invariants as k6 = i s.. s.. (12U) 2 1k ki J. - 1 ^ s.. s. s . (125) J3 " 3 3 lk km mi The invariants of stress and stress deviation tensors can he related by using Equation 122. In view of Equations 122 and 118, T. 1 ... s. . lk ki 1 \ a - - J.Ô.. lia.. • lk 3 1 ikf l ki - 3 Ji 5ki 2 t . ' 0 0 — — Jo 0 + —4 6. 6, . (126) lk ki 3 1 lk ki 9 ik ki 2J^ 5 Equation 126 Since 6ikôki = 3 » aik6ki ” Ji » and aikaki becomes j, - j I j2 (127) J2 " J2 6 J1 Similarly, it can he shown that (128) J3 - J 3 - I J1J2 + If J1 Principal stresses 38. The three principal values of stress tensor are referred to as principal stresses and are denoted hy (using the principal directions as reference axes) a 0 0 [a] 0 a2 0 (1 2 9 ) 0 0 o3 It should he pointed out that the ordering of the principal stresses in Equation 129 does not imply that the numerical value of is greater than a^ or • As discussed in Part II, the three principal values are the roots of the characteristic equation of the tensor À3 - I X2 + II À - n i 0 a a a (130) where I = J = a 1 nn (131a) °22a23 ailü13 °llal2 II = + + (1311)) a32a33 a31°33 °2ia22 III -M (131c) The two coefficients 11^ and III^ are usually denoted by and , respectively, and can be related to , and by using Equations 107, 110, 118, 119, and 120. J2 - IXa = I (J1 - 2J2) (132) (133) J3 = IITo * J3 - J2J1 + ? The invariants of stress deviation tensor can also be expressed in terms of , Jg , and • In view of Equations 127, 128, 132, and 133, we obtain I J 2 T 131 3 1 " J2 ( +) 1 2 3 = - J3 3" J1J2 + 27 J1 (135) Principal stress space and octahedral stresses 39* Since the three principal stresses are orthogonal, they form a three-dimensional space called the principal stress space (Figure 5). Of particular interest in the principal stress space are the octahedral planes. The direction cosines of a normal to an octahedral plane are (Figure 5) U8 N cos (N , a1 ) = cos (N 9 a^) = cos (N , a^) = cos (5*+° b k ' ) JL (13 6 ) /3 The normal and shear stresses on octahedral planes are denoted as Qoct an^ Toct 5 respectively. The magnitude of °OQ^ and Toct can he determined from the transformation law of stress tensor^ a 1 0 0 0 a2 0 ( 137 ) oct 3 (°1 + a2 + 03 > 0 0 a. k9 and oct 3 >/ (ax - a2)2 + (ax - ag)2 + ( Using Equations 12^ and 118 it can be shown that for a general state of stress 0oct 3 (139) 1 2 ~ oct V 3 J 2 (iho) Equations 139 and lUo indicate that the octahedral stresses are also invariant. The octahedral space (t^ versus ) is commonly used for plotting stress paths for various laboratory tests. In this report we will use V 3/2 versus CQct space (i.e., versus J ± /3) for defining stress paths. Examples of sim- ple states of stress i+O. The following states of stress are often utilized in the laboratory in order to determine the stress-strain properties of a material: a,. Spherical or hydrostatic state of stress. 50 b. Uniaxial state of stress, i o o o i—i CD a. . = 0 0 0 ij 0 0 0 U c. Cylindrical state of stress. a 0 0 i ; 1 0 c30 ij 0 0 a. ° 2 = °3 d. Triaxial state of stress. a 0 0 á a. . = 0 a2 0 10 0 0 a, w e. Pure shear. 0 o12 o a. . = ij °21 0 ° 0 0 0 Note that in examples a through cl all stresses are principal stresses. Stress paths associated with the above states of stress can be readily defined in the versus J^/3 space. The stress path associated with spherical or hydrostatic state of stress is shown in Figure 6a. It is noted that for spherical state of stress is zero. The stress path associated with uniaxial state of stress is shown in Figure 6b. For uniaxial state of stress = a //3 and <1^/3 = a1/3 result- ing in the expression /3 (J^/3) for the stress path. 51 a. SPHERICAL OR HYDROSTATIC STATE OF STRESS c. CYLINDRICAL STATE OF STRESS d. CYLINDRICAL STATE OF STRESS (CONSTANT Jj/3) (CONSTANT ii h 3 e. PURE SHEAR Figure 6. Stress paths associated with simple states of stress 52 Figures 6c and 6d depict special stress paths associated with cylindrical state of stress. In Figure 6c the material is first loaded hydro statically and then sheared while J^/3 is kept constant. In Figure 6d the material is first loaded hydrostatically and then sheared by in creasing while keeping constant. Since for cylindrical state of stress = (a^ - a^)/-/3 and J^/3 = (a^ + 2o^)/3 it follows that the expression for the stress path of Figure 6d becomes - o^j . The stress path associated with pure shear test is shown in Figure 6e. In the case of pure shear J^/3 = 0 ! In the actual laboratory coordinate system, the stress components and = o^ associated with cylindrical state of stress are usually de- noted by o (axial stress) and a = a. (radial stress), respectively, a r u In the case of triaxial state of stress, the stress components , a , and c are denoted by a , a , and a , respectively. For <— O' x y z pure shear the only nonzero stress component a i s generally denoted by t . Strain Tensor Ul. Let us consider a cylindrical specimen of length and extend it to length £ . The ratio £/£^ is defined as the stretch X . A = % / l Q (lUl) The question is, what is the axial strain in the specimen? There are several measures of strain that can be used to determine the axial 3 strain e in the specimen. These measures, named after Cauchy, Green, Hencky, Almansi, and Swainger, respectively, are: 1 (li|2a) (lU2b) 53 e H = InX 0 , (lU2c) A £ (ll+2d) S e (lU2e) In order to demonstrate the difference between the various measures of strain given in Equation lU2, let 1 = 2i^ (i.e., let the length of specimen be doubled). The stretch X = 2 in such case and from Equa tion lU2 it follows that c e = 100s? ' G e = 15 0 $ H £ = 69% (1U3) A e = 3 7 . 55? S £ = 50% . As observed from Equation 1 U3 , for a stretch of X = 2 , the difference between the various measures of strain is quite appreciable. Now let 1 = 1.25&Q , which gives a stretch of X = 1.25 In view of Equa tion 11+2, for X = 1.25 the various measures of strain become c e = 25% G e = 2QI H £ = 22% (iM O A OO £ = H S £ = 20$ It is observed in this case that the difference between various measures of strain is not as appreciable as was the case for X = 2 . If the 5b stretch X is further reduced, say X = 1.1 , Equation lk2 -will result in £C = 10% eG = 10.5$ ,H = 9% (1U5) eA = 8.7% eS = Q% Therefore, for small deformations (infinitesimal strain theory) the various measures of strain will yield approximately the same results. Our interest here is also within the framework of infinitesimal strain theory and we adopt the Cauchy measure of strain for further analysis. 1+2. In order to determine strain-displacement relations and de fine the infinitesimal strain tensor, we consider a particle P with position vector in the x^ coordinate system as shown in Figure 7 Figure 7* Particle displacement in x^ coordinate system 55 We assume that the particle undergoes displacement u^ and assumes a new position vector as depicted in Figure 7* From Figure 7 we can write x. + u. x. (iU6) 1 i i or u. x. X. l i 1 dvr) Since x. is a function of x. , i.e., x. = x.(x.) we can differen- 1 1 _ i 1 J tiate Equation lh6 with respect to x^ ; thus 6.. + u. . = x. . (lU8) ij The terms x. . and u. . are called the coordinate gradient and dis- placement gradient matrices, respectively. The displacement gradient matrix can he expressed as the sum of a symmetrical system and a skew- symmetrical system (see Equation 15) u (u. . + u. .) + ~ (u. . - u. .) (1U9) J»1 2 i,J The first term in Equation 1U9 is symmetrical and is called the infini tesimal strain tensor e .. ; thus, e . . + u . .) (150) ij J The second term in Equation 1U9 is skew-symmetric and is called the rotation tensor . ; thus, ft. . = t: (u. . - u. . ] (151) ij 2 i,j J,i Equation 150 relates the components of infinitesimal strain tensor with components of displacement vector. 56 1*3. To demonstrate the application of Equation 150, consider a rod of length £Q extended to length £ as shown in Figure 8. The boundary conditions associated with displacement u_^ in the direction are u = 0 at xn = 0 (152) = £ - £q at x^ = £, For a homogeneous state of strain to exist in the rod, the displacement u^ must he a linear function of x^ - Thus, (153) ui = Cxi where C is a constant. In view of Equation 152, Equation 153 becomes £ - £, 0 x„ (15U) U1 = £ 0 Substituting Equation 15^ into Equation 150 we obtain £ - £ 0 ( ^ , 1 + u^ ) (155) 'll 2 £ 0 57 which is the Cauchy measure of strain (see Equation 1^2a). Invariants of strain tensor UU. Strain tensor is a second-order tensor and obeys the trans formation law given in Equation i.e., e! . = a. a. e (156) i j m jm run where e! . is referred to the x! coordinate system. There are, ij 1 therefore, three independent invariants associated with the strain tensor. As for the invariants of stress tensor, we define the invari ants of strain tensor as I I e 1 e nn (157) 1 I — £ £ (158) 2 2 ik ki 1 I (159) 3 3 eik£km£mi Strain deviation tensor ^5. Strain tensor can he expressed as the sum of two symmetric tensors in the following manner = E. . + £ ó . . (160) ij ij 3 nn ij where the tensor 1 E. . £ £ 6 . . (161) ij 1J. . 3 nn ij is referred to as the strain deviation tensor and e <5../3 is called nn ij the spherical strain tensor. As for invariants of stress deviation tensor, we define the invariants of strain deviation tensor as I 2 - 1 EiA i (162) 58 Il = i HI™ = i E._ E E . 3 3 E 3 ik km mi (163) The invariants of strain deviation tensor can also be expressed in terms of the invariants of strain tensor as follows: I » = I _ i X2 X2 6 I1 (16U) — 2 — 23 J 3 X3 - 3 Vs + 27 (165) Principal strains k6. The three principal values of strain tensor are referred to as principal strains and are denoted by (using the principal directions as reference axes) 0 0 "I [e] = 0 0 £2 (166) 0 0 £. The principal strains are the roots of the characteristic equation of strain tensor X3 - I X2 + II X - III = 0 £ £ £ ( 167) where I £ = i 1 = £ nn (l68a) £ £ e22 e23 £11 e13 11 12 II = + + e (l68b) U) U) 00 00 CO 1—1 £ £ £32 e33 21 22 III = | e I (l68c) 59 The two coefficients II and III are denoted by I and I , e e 3_. respectively, and are related to the invariants 1^ , > and I^ as follows: I (169) 2 I3 = rII£ * h - V i + ì Tì (170) The invariants of strain deviation tensor can also be expressed in terms of 1^ , Ig , and I^ : I' = — I2 - I (171) 2 3 1 2 — 1 2 3 I'=I - — II + — IJ (172) 3 3 3 1 2 27 1 Examples of simple states of deformation 1+7. The following states of deformation are often utilized in the laboratory in order to determine the stress-strain properties of the material: a. Uniform dilatation. e. . ij b. Uniaxial state of strain. 60 £n 0 0 ‘ 1 £. . = 0 0 0 1J 0 0 0 c. Cylindrical state of strain ( = e3). £2 = 1 £- 0 0 1 £. . — 0 0 ij 2 0 0 e 2 d. Triaxial state of strain. 0 0 1 / 0 0 2 0 0 £ e. Simple shearing deformation (no volume change). 0 0 el2 0 0 e21 0 0 0 Note that in examples a through d all strains are principal strains. Strain-Rate Tensor 1+8. The time derivative of infinitesimal strain tensor is re ferred to as rate of infinitesimal strain tensor, or simply strain-rate tensor, e.. ; thus, d ( \ (173) ij dt ^ij) 61 where cL/dt indicates differentiation with respect to time. In view of Equation 150, the strain-rate tensor takes the form e. . = 7T1 ( (v. . + v. .) (17U) ij 2 i,j where v^ = components of velocity vector. Invariants of strain-rate tensor U9 . Strain-rate tensor is a second-order symmetric tensor and, like the stress and strain tensors, it obeys the transformation law given in Equation 9^- Similarly, we define the invariants of strain- rate tensor as I = I. = é 1 e nn (175) — i — i I0 = ~ II. = ~ é.. L . 2 2 e 2 lk ki (176) h = I nií = i (177) Strain-rate deviation tensor 50. Strain-rate tensor can he expressed as the sum of two sym metric tensors in the following manner e. . e 6. . (178) ij nn ij where the tensor (179) É. . £ . . 6 . . ij 1 J nn ij is called the strain-rate deviation tensor and e S../3 is called nn ij the spherical strain-rate tensor. We define the invariants of strain- rate deviation tensor as 62 — 1 — 1 • II = t IL = ± E.. E. . 2 2 E 2 lk ki (180) J L *1 ------“I • T t - — T T T = — E E E . (181) x3 " 3 111^ 3 ik km mi The invariants of strain-rate deviation tensor can also be expressed in terms of the invariants of strain-rate tensor: 1 = In - (182) T2 z 2 2_ I + (183) 3 3 27 Principal rates of strain 51. The three principal values of strain-rate tensor are denoted ky ) 0 0 &1 0 0 (1810 *2 0 0 £. and are called the principal rates of strain. The principal rates of strain are the roots of the characteristic equation of%strain-rate tensor X3 - I.x2 + II.X - III. = 0 £ £ £ (185) where I. = In £ (l86a) £ nn 63 # m • • e22 e23 ell e13 11 £12 II. + + (186b) e • . £ £ £ £ 32 e33 31 33 e21 e22 III. (l86c) £ The two coefficients II. and III. are denoted by IG and Io , re £ £ JL ^ spectively, and are related to the invariants 1^ , ^2 9 an<^ ^3 as follows: II. (187) £ = III. = l0 - I^I (188) e 3 2 1 The invariants of strain-rate deviation tensor can also he expressed in . . * terms of 1^ , 1^ , and 1^ : f. = M - I, (189) 2 3 1 —I'=I • - — 1 II • . + — 2 I • (190) 3 3 3 1 2 2T 1 Equations of Continuity and Motion 52. The motion of any continuum is governed hy the following laws a_. Conservation of mass, b. Conservation of energy. c_. Balance of linear momentum, cl. Balance of angular momentum. _e. Principle of inadmissibility of decreasing entropy. 61 These lavs constitute the basic axioms of continuum mechanics. In the absence of distributed couples, the balance of angular momentum leads to the symmetry of stress tensor, a . . = a.. . If mechanical energy is the only form of energy to be considered in a problem (as is the case in this report), the above principles lead to the continuity equation ft + (PTi),i “ 0 <191) and the equations of motion a. . . + f. = pa. (192) 1 l vhere p = mass density , v^ = components of velocity vector , f^ = components of body force , and a^ = components of acceleration vector . Equations 191 and 192 are applicable to all materials. Constitutive Equations 53. Equations 191 and 192 constitute four equations that involve ten unknown functions of time and space: the mass density p , the three velocity components v^ , and the six independent stress com ponents a. . . The body force components f^ are known quantities and the acceleration components a^ are expressible in terms of the veloc ity components v_^ . Obviously, Equations 191 and 192 are inadequate to determine the motion or deformation of a medium subjected to ex ternal disturbances, such as surface forces. Therefore, six additional equations relating the ten unknown variables p , v. , and a. . are required. Such relationships are referred to as constitutive equations, which relate the stress tensor a . . to deformation or motion of the ij medium. As was pointed out previously, Equations 191 and 192 are applicable to all materials, whereas constitutive equations represent the intrinsic response of a particular material. Furthermore, a con stitutive equation provides a mathematical description or definition of an ideal material rather than a statement of a universal law. The 65 general form of a constitutive equation may be expressed by the func tional form (considering only mechanical effects) 9 p) = 0 (193) or (19*0 where d ^ = time derivative of stress tensor . Equations 191, 1 9 2 , and 19^ (or Equation 193) , therefore, constitute ten equations in ten un knowns and will lead, in conjunction with kinematic relations given by Equations 150 and 17^-, to a complete description of the boundary- value problem. In addition to the above-mentioned equations, boundary conditions in terms of boundary displacement and/or surface forces must also be specified to completely define a particular problem of interest. 5^. In order for constitutive equations to describe physical ma terials adequately, the functional forms f .. or g . . must remain ij invariant with respect to rigid motion of spatial coordinate . This requirement stems from the fact that the response of a material is independent of the motion of the observer. Furthermore, the functionals f.. or g.. must be consistent with the general principles of con- i J i J servation or balance of mass, momentum, and energy. 55. We adopt Equation 19^, relating four second-order symmetric tensors £^s , , and d ^ , as a "basis for development of various constitutive equations in the following parts of this report. 66 PART IV: CONSTITUTIVE EQUATIONS OF ELASTIC MATERIALS 56. For an elastic material, the state of stress is a function of the current state of strain only. Furthermore, an elastic material re turns to its initial state after a load-unload cycle of deformation (no permanent strain). The stress tensor can, therefore, be expressed in terms of strain tensor a. . F..(e ) (195) ij lj mn where F ^ = elastic response function. Two different procedures have been utilized in order to determine the response function F^ for iso tropic materials. The first procedure, referred to as Cauchy’s method, is based on the Cayley-Hamilton theorem (Equation 59)* The second pro cedure, referred to as Green’s method, is based on conservation of energy. Both of these methods are dealt with in this part of the report. Cauchy’s Method 57* The response function F.. in Equation 195 can be expanded as a polynomial in the strain tensor e. . i. e. , iJ a. . a^ + a_, e. . + a^e. £ . a0e. e e . + (196) 13 0 1 ij 2 im mj 3 im mn nj where aQ » a^ , ... a^ are real coefficients. Utilizing the Cayley- Hamilton theorem we can express Equation 196 in the following form (see Equations 62 and 63) a. . 67 e . . W.Ô. . Y a. . + ^0a. a . (198) ij 0 ij 1 iJ 2 in mj where ’ anà are elas'ti-c response coefficients which are polynomial functions of stress invariants. From Equation 197 it follows that for isotropic elastic materials the initial state of stress is hydrostatic, i.e.-, a . . Also, using the transformation law of a second-order tensor (Equa tion 9*0, it can he shown that Equation 197 is form invariant with re spect to rigid motion of a spatial coordinate system, i.e., crf = a .a .a. . mn mi nj ij — p ô T + (p e ' + where £mn is referred to the Primed (rotated) coordinate system. We can now utilize Equation 197 to develop various types of isotropic elastic constitutive equations. Linear elastic material 58. For linear elastic materials the response coefficient ( p ^ vanishes. The response coefficient cp1 is a constant and < p Q is a linear function of the first strain invariant. Assuming that the initial state of stress is zero, the constitutive equation of linear elastic ma terial can he written as a .. = AI_ 6.. + Be. . (201) ij 1 ij ij where A and B are material constants. In order to determine the physical meaning of the material constants A and B let us consider a simple shearing deformation defined by 68 '0 0 * el2 e . . = 0 0 (2 0 2 ) ij e21 0 0 0 For this state of deformation, Equation 201 reduces to 0 0 Bei2 a. . = 0 0 (203) ij Be21 0 0 0 Since e1 2 ' is half the shearing strain (see Equation 150), it follows that B is two times the shear modulus which we define as G ; thus, B = 2G (201+) Next, we consider uniform dilatation defined by e. . (205) ij For this state of deformation, Equation 201 becomes (invoking Equa tion 20U) a. . AIn + 2G (206) ij Taking the trace of (let i = j ) we obtain ij 69 a.. 11 I 3 ?) 1 (207) Equation 207 relates pressure (J^/3) to volumetric strain (i ). The slope of the pressure-volumetric strain relation is defined as bulk mod ulus K ; thus, 2G A + = K (208) or _ 2G A = 3 (209) The constant A is usually denoted as A and is referred to as the Lame constant. In terms of the shear and bulk moduli, the constitutive equation of linear elastic material (Equation 201) then becomes a . , K L , < 5 . . + 2G e. . (210) ij 1 13 \ iJ The expression e.. - I, <5../3 is recognized as the strain deviation 1 J 1 1 J tensor E . . (Equation l6l); thus, l J a. . KI 6. . + 2GE.. (211) ij 1 13 13 From Equations 122, 207, and 208, it follows that the constitutive equa tion of linear elastic materials can also be written as Sij = 2GEij <212a) — = KI1 - (212b) Equation 212 indicates that for linear elastic materials volumetric strain is caused by hydrostatic stress only, and that the shearing re sponse of the material is independent of pressure. 70 59- Using Equation 212 we can readily express the strain tensor in terms of stress tensor S. . E = — ij 2G (213a) X (213b) 1 3K or, using Equation l6l, J s (2lU) eij = 9K 6ij + 2G 60. We will now proceed to examine the behavior of linear elastic materials under various states of stress and deformation. Let us first consider uniaxial state of stress, a common laboratory test, defined by 0 0 °1 a, . = 0 0 0 (215) ij 0 0 0 For this state of stress, Equation 2lb results in Cis-h/3K + G\ 0 (2G - 3K\ e . . = (216) 1 3 V i 8kg )c /2G - 3K\ ( i 8kg )c Equation 216 indicates that under uniaxial state of stress 9KG (J = — ------P ( 217) 1 3K + G 1 71 '3K - 2G £ £ (218) 2 6K + 2G)1 The ratio cr^/e^ un&er uniaxial state of stress is referred to as Young’s modulus E , and the ratio of radial strain to axial strain is called Poisson’s ratio v ; thus (for an incompressible elastic material v = 1/2) = 9KG 3K + G (219) = 3K - 2G (220) V 6K + 2G 6l. Another common laboratory test is the uniaxial strain test defined by 0 0 £1 0 0 0 (221) 0 0 0 For this state of deformation, Equation 211 results in (K + W 3 ) e 1 0 0 a. . 0 (K - 2G/3)e1 0 (222) ij 0 0 (K - 2 G /3 )c1 From Equations 222, 219, and 220 it follows that under uniaxial strain condition a! + (i + v )(l - 2 v ) £1 (223) /3K - 2G\_ _ ( v \ (22U) a2 = a 3 V3K + bG)al ~ \1 - v)C 72 It is noted that is the radial stress required to prevent radial strain. The ratio cr^/e^ un(^-er uniaxial state of strain is referred to as the constrained modulus M ; thus, (225) M - K + 1,G/3 = (1 gv) Using Equations 223 and 22^ we can determine an expression for the stress path associated with the uniaxial state of strain in the versus J^/3 space: (226) In terms of Poisson’s ratio v , the equation of the stress path becomes vtt = v 3 <227) 62. Next, let us consider the behavior of linear elastic mate rials under condition of plane strain defined by 0 ell £12 e. . = 0 (228 ) ij £21 £22 0 0 0 For condition of plane strain, Equation 211 results in — G)e 2Gs 3 / h 12 2GX K- 3 )'22 2Ge (229 ) ij 21 (* - f> u + (K + I ° > '22 (K - f ) ( Ell + e22) 73 From Equation 229 it follows that _ /3K - 2G\/ x 230 a33 \6K + 2G/^ail ff22 v(c,ll + 02 2 ) ( ) Equation 230 gives the magnitude of stress a^ necessary to maintain plane strain condition, i.e., e^ = 0 . 63. The counterpart of plane strain is the condition of plane stress defined by 0 CT11 °12 a. . = 0 (231) ij °21 a22 0 0 0_ don 2lk results in 12 9K + 3g)°11 2G ♦(k - y 22 21 e. . (232) ij 2G ( 9K “ 6g)CT11 + (fe+ k ) 22 (9 K “ 6g)^ CT11 + °22^ From Equation 232 it follows that (233) '33 (9K ~ 6G)^ail + 022^ 1 - v ^Ell + £22^ Equation 233 gives the magnitude of strain e^ produced by condition of plane stress, i.e., a^ = 0 . 6k. The constitutive equations of linear elastic material ex pressed in terms of various combinations of elastic constants are given in Table 1 for ready use. Table 1 Constitutive Equations of Linear Elastic material Expressed in Terms of Various Combinations of Elastic Constants Elastic Constants Young * s Modulus» E ______Poissonts Ratio» v______Shear Modulus, G on , GE — 2G T x E aiJ ~ (1 + v)(l - 2v) Zl6iJ 1 + v ciJ °ij = 2Geij 3G - E V i J - 1 + V ^ V T X eij " E aiJ ” E Jl6ij eiJ = 2G^ “ (2G " E)Jl6ij 2Gv . V °ij = 2GeiJ 1 - 2v X1 lj \J1 6KE 9K2 - 3KE = 3vK 3K(1 - 2v) Bulk a. J = 9K - E eiJ 9K - E 'ij ~ 1 + v 1 + v ’a * + (K - r X * » Modulus, au V iJ K _ 9K - E „ 3K - E T . ij “ ¿KE ij " 6KE Jl6ij Nonlinear elastic material 65* Constitutive equations for various classes of nonlinear elastic material can be developed from the general form of the Cauchy- elastic constitutive equation (Equations 197 and 198). Before we de velop constitutive equations for various classes of nonlinear elastic material it would be beneficial to examine the significance of the second-order terms . and a. a . in Equations 197 and 198. Con- e.1 m emj 1m mj sider a simple shearing deformation of amount 2y defined by the follow ing strain tensor 0 Y 0 e . . = Y 0 0 (23M ij 0 0 0 For this state of deformation, Equation 197 results in the following ex pression for the stress tensor " 2 1 0 0 0 Y 0 Y 0 0 - + 0 2 1—1 a. . = $ 0 1 0 - Y 0 0 0 Y 0 (2 3 5 ) ij 0 + *2 0 0 1 0 0 0 0 0 0 - _ From Equation 235 5 the shearing stress and shearing strain are related by (236) "12 = +1T and the normal stresses are given as ^ ° (237a) all °22 ^0 + 2Y (237b) °33 *0 Equation 237 indicates tfeat to maintain a simple shearing deformation (Equation 23*0, normal stresses must be applied to the boundaries of the specimen. Since two of the normal stresses are unequal, Equation 237 predicts the occurrence of normal deviatoric stresses S11 = S22 = 3 (p2y2 (238) on the shearing planes. This is a direct consequence of the second- order term e. e . in Equation 197 and is a departure from the linear lm mj theory where ^ = 0 . We now consider the counterpart of simple shear ing deformation, i.e., simple shearing stress, and show that Equa tion 198 will predict volume change for this state of stress. Consider a simple shearing stress of amount t defined by the following stress tensor 0 T 0 T 0 0 (2 3 9 ) 0 0 0 For this state of stress, Equation 198 results in the following expres sion for the strain tensor ' g 2 1 H O 10 0 O TOO ro H O 0 10 TOO 0 (2U0) + h + *2 O H O 0 0 0 0. ■5“ (0 From Equation 2^0 it follows that e. . 3¥, + 2V2t (2hl) 11 h = which indicates that simple shearing stress is accompanied by volume change. Also, from Equation 2^0 it follows that there are normal devia toric strains 77 1 'PgT (242) E11 = E22 3 associated with the volume change. The occurrence of deviatoric strains is a direct consequence of the second-order term a. a . in Equa- im mj tion 198. From these two examples we can conclude that in the case of nonlinear elastic materials volumetric strains are caused by both the hydrostatic and shearing stresses. Also, the shearing response of the material is dependent on the hydrostatic state of stress. In order to further demonstrate these coupling effects we consider a combined state of hydrostatic and simple shearing stress given by the following stress tensor p T 0 T P 0 (2U3 ) 0 0 p where P = superimposed hydrostatic stress. For this state of stress, Equation 198 results in the following expression for the strain tensor “ “ “ -j-,2 , 2 1 0 0 p T 0 P + T 2Px 0 2 2 e . . = T. 0 1 0 T p 0 2Pt P + T 0 (244) ij 0 + h + *2 0 0 1 0 0 p 0 0 P2 , _ The volumetric strain I then becomes e^i = I = 3¥0 + 3^P + ¥2(3P2 + 2t2 ) (21+5) and the shearing strain e 2 takes the form (246) E12 = V + 2V2Pt Equations 245 and 246 once again illustrate the coupling which exists 78 between the hydrostatic stress (or volumetric response) and the shearing response of nonlinear elastic materials. 66. We will now proceed to develop constitutive equations for various classes of nonlinear elastic material within the framework of the Cauchy elastic constitutive equation (Equations 197 and 198). The simplest form of nonlinear elastic material is the second-order stress- strain relation where terms in strain up to the second power are re tained in the stress-strain relationship. To derive the constitutive relationship for second-order elastic material we will start from Equa tion 197 and express the response coefficients (247a) *0 = Vi+ Vi+ Vo *1 - C1,+ Vi (24Tb) a. . [ + (2G + + C^e. . (248) 6 lm emj 67. Equation 2^8 contains six material constants. The physical meaning of these constants and the manner in which they can be deter mined from laboratory test results can be demonstrated by examining the behavior of second-order elastic materials under various states of stress and deformation. Let us first consider a simple shearing 79 deformation defined by Equation 202. For this state of deformation — 2 1.^ = 0 and = £-^2 » an(^ Equation 2^8 results in the following rela tions for the components of stress tensor 012 = 2Ge12 (2l+9a) 011 = °22 = ^C3 + C6^£12 (2l+9b) (2U9c ) 033 = C3el2 Equation 2*+9a indicates that a second-order elastic stress-strain rela tionship predicts a linear relation between shearing stress and shearing strain. Equations 2^9b and 2^9c give the magnitude of normal stresses, as a function of shearing strain, required to maintain shearing deforma tion. It is noted that the normal stresses are not uniform, thus re sulting in normal deviatoric stresses S S = 1 c e2 11 22 3 °6 12 (250) on the shearing planes. The significance of the material constants and the combination (C^ + Cg) is realized from Equations 2*+9c and 2*+9b, respectively. 68. We next consider uniform dilatation defined by Equation 205. For this state of deformation 1 ^ = 1^/6 and Equation 2^8 results in the following relationship between pressure and volumetric strain 3 = K I 1 + (C 2 + H * 3 C5 + 9C6)I1 <251) Equation 251 describes a parabolic stress-strain relationship. The material constants K and the combination ^C2 + C^/6 + C^/3 + Cg/9^ can be determined from experimental data by curve fitting techniques. It is noted that K is the initial slope of the pressure-volumetric strain curve and is a positive constant. If the combination of the ma terial constants in the parentheses is also positive, the stress-strain curve will be concave to the stress axis. If, on the other hand, this 8o combination is negative, the stress-strain curve -will be concave to the strain axis. 69• We will next consider uniaxial state of strain defined by _ 2 Equation 221. For this state of deformation 1^ = and = e^/2 and equation 2^+8 results in = (k + I g)6i *(o2 * | c3 + c5 + c 6 ± (252 ) a1 y (253) °2 = ’ 3 = (K - 1 GK + (C2 + r ) ei Equations 252 and 253 also describe parabolic stress-strain relation ships. The combinations of material constants' + C^/2 + ^5 + ^g) and (C2 + C /2) can be determined from experimental data by curve fit ting techniques. As was pointed out previously, the shape of the stress- strain curves predicted by Equations 252 and 253 depends on the sign of the combination of the material constants in the parentheses. The stress-strain curves will be concave to the stress axis if these combina tions are positive, and concave to the strain axis if they are negative. Since is the only nonvanishing strain component in uniaxial strain configuration, we can use Equations 252 and 253 to relate stress dif ference to strain difference , i.e., a 1 - a2 = 2Gex + (C5 + Cg)e^ (25*0 Again, the shape of the stress-strain curve is determined from the sign of the combination (Cj_ + Cg) of the material constants. 70. More complicated states of stress and deformation, such as plane strain and triaxial stress conditions, can also be studied within the framework of second-order stress-strain law. Such states of stress generally lead to lengthy mathematical expressions between stress and strain components. For example, consider cylindrical state of strain defined by 81 e 0 0 1 e . . 0 e2 0 (255) ij 0 0 For this state of deformation = e1 + 2e2 and = 1 / 2 ^ + 2e2j and Equation 2^8 results in the following expressions for the components of stress tensor 01 = (K + I ° k + (C2 + T- * c5 * cô)el + (2K - I G) £2 + K + c3) 4 + K + 2c5) £i £2 <256) °2 ' °3 ” (K ' I G) £l + (C2 + f ) e1 ♦ (2K ♦ f ) c 2 * (^a + °3 + 2c5 + ce)E2 + (^a * c5)£i £2 (257) Various stress paths may be employed in a laboratory test maintaining a cylindrical state of strain. The most common stress path used with this state of strain is to keep the lateral stress constant while in creasing (Figure 6d). For this stress path it is possible, in prin ciple, to solve for &2 in term of , using Equation 257 (since ct^ is a constant), and then substitute the resulting expression into Equa tion 256 to develop a relationship between and . Other stress paths such as constant J /3 path (Figure 6c) and uniaxial stress test (Figure 6b) can also be considered. 71. Following the same procedure we can develop and analyze more complicated nonlinear elastic stress-strain laws. Let us consider a third-order law where terms in strain up to the third power are retained in the stress-strain relationships. Accordingly, the response coeffi cients 82 (K-f g) (258a) h + C2T1 + C3I2 + v ? + C8I1I2 + V 3 = 2G + (258b) C5Il + C1QI1 + (258c) *2 - C6 + C12I1 where through C_^2 are six additional material constants which must he determined experimentally. A third-order elastic stress-strain law formulated within the framework of Equation 197 (Cauchy’s method), therefore, contains twelve material constants. In view of Equation 197 the constitutive equation of third-order elastic material becomes h * V l + c312 + V l * C8I1I 2 + V s i]5ij 2G ♦ ♦ C104 + CU I2>)e . . + (Cc + C, 0I, )e. e . (259) / iJ 6 12 r 1 1 mj It is noted that if third-order terms in strain are neglected, Equa tion 259 reduces to Equation 2^8 (constitutive equation of second-order elastic materials). It was pointed out previously that a second-order elastic stress-strain relationship predicts a linear relation between shearing stress and shearing strain. Nonlinear relation between shear ing stress and shearing strain is due to third- or higher-order terms in strain tensor. This phenomenon, which is a departiare from second-order effect (see Equation 2^9a), can be demonstrated by examining the behav ior of third-order elastic materials under simple shearing deformation defined by Equation 202. For this state of deformation 1^ = 0 , _ 2 — I2 = 9 an(3- I3 = 0 and Equation 259 results in the following re lation for the shearing stress cr ^ C e3 (260) °12 = 2GE12 + 11 12 Equation 260 is a third-order equation in shearing strain . The behavior of third-order stress-strain law under various states of stress 83 and deformation can also be studied similar to the second-order law. In the next section we will consider less complicated and perhaps more useful forms of elastic stress-strain laws referred to as quasi-linear elastic material. Quasi-linear elastic material 72. Nonlinear elastic stress-strain laws are too complicated for application in all engineering problems. In many engineering problems only an approximate or gross behavior of the material under considera tion needs to be modeled. For this reason, we will develop a number of simple stress-strain laws for simulating the gross behavior of a number of materials. We will start with Equation 197 (or Equation 198) by making the assumption that the response coefficient cr. . 4>n6. . + <)>,£.. (261) ij 0 1J 1 IJ where, as before, J 1 (262a) 3 l1! S. . (262b) ij 1 ij In view of Equations 12*+, 162, and 262 we can write ♦l = (263a) > 40 (263b) 81+ Substituting Equation 263 in Equation 26l results in the following gen eral constitutive equation for quasi-linear elastic material a. . (2610 ij It is only necessary to postulate (based on experimental evidence) math ematical expressions for and in terms of strain invariants in order to utilize Equation 26^ for any material of interest. The inverse of Equation 26^, resulting in strain-stress law, can be obtained from Equation 198 by assuming that i-s zero and following the above pro cedure. The resulting relationship becomes e. . (265) ij To use Equation 265 we need to express I and 1^ in terms of stress invariants. For example, in the case of the linear elastic materials (266a ) J 1 KI, (266b) 3 and Equations 26 k and 265 reduce to constitutive equations of linear elastic materials (Equations 210 and 21^-, respectively). 73. We will now proceed to develop constitutive equations for various classes of quasi-linear elastic materials which are of interest for engineering application. For the simplest class of quasi-linear elastic material, in which there are no couplings between the deviatoric and volumetric responses of the material, we can write (267a) (267b) 85 where the functions f^ and f^ must be determined based on experi mental evidence. A good approximation for a number of materials (such as clay soils) is to assume the following relations for f^ and f (268a) + V B (268b) where P , a , k , and k0 are material constants which must be de- termined experimentally. Equations 268a and 268b are depicted graphi cally in Figure 9* It is observed from Figure 9 that defines an initial hydrostatic state of stress (for materials that can sustain ten sion, P can be taken to be zero) and P defines the maximum hydro- u b static tensile stress that the material can sustain before it fails (breaks) at such a tension. For materials that cannot sustain tension, the material constant P^ is zero. In this case P defines the state b 0 of "ease” or the initial stress state of the material. The material constant k^ is proportional to the inverse of the initial shear modu lus and k^ is the inverse of the ultimate shear strength of the mate rial. Substituting Equations 268a and 268b in Equation 26^ results in the following quasi-linear elastic constitutive equation a. . 8. . (269) ij + PB} 'B ij eij The inverse of Equation 269 resulting in strain-stress law can be ob tained by inverting Equations 268a and 268b and substituting the result- ing expressions for I and y[*l into1 Equation 265 as follows: 1 e. . £n V 3 * PB a. . (270) ij 3 , P0 * PB , ij 86 Vi iue9 Asmdrltosis forquasi-linear relationships Assumed 9-Figure TENSION (-) | COMPRESSION (+) a. PRESSURE-VOLUMETRIC STRAIN RESPONSE (EQ.268a)RESPONSE STRAIN PRESSURE-VOLUMETRIC a. b. DEVIATORIC RESPONSE (EQ. 268b)RESPONSE DEVIATORIC b. elasticmaterial Ji 3 87 7^. Equation 269 (or Equation 270) is a simple but useful consti tutive equation which can be used to study the stress-strain behavior of a number of physically nonlinear materials. It contains only five mate rial constants which have physical meaning and can easily be determined experimentally. Having determined the numerical values of these mate rial constants we can use Equations 269 and 270 to predict the behavior of the material under any state of stress and deformation. For example, consider condition of uniaxial strain defined by Equation 221. For this — 2 state of deformation 1^ = and 1^ = e^/3 , and Equation 269 results in 2 aen 3 £1 a. = (271) (p0 + PB )e PB + k l + — e i 1 /3 1 1 3 £1 (272) °2 = °3 = (p0 + PB)e - PB - k + — e 1 /3 1 Equations 271 and 272 predict stress-strain curves that may initially be concave to the strain axis and then become concave to the stress axis as the vertical strain increases. Using Equations 271 and 272 we can determine the stress path associated with the state of uniaxial strain in the n versus J /3 space, i.e., + P. B — £n a P 0 + PB (273) 3 + PB /3 k_ + — £n 1 a P0 + PB Next consider the behavior of the material under condition of uniaxial stress defined by Equation 215 (assuming that P = 0). For this state — 2 L of stress J = a and = cr /3 and Equation 270 results in 88 + P. B — £n f V i {2.1k) £1 3 a 'B 1 - — a ✓3 + P. B — £n i V i (275) £2 = 3 a 'B 1 - — CTn /3 75. A more complicated quasi-linear elastic material model can now tie constructed by assuming that the shearing response of the mate rial is a function of both the hydrostatic and the deviatoric stresses, while the volumetric response is only a function of pressure. Accord ingly, Equation 267a is still valid while instead of Equation 267b we can write = f 2 ’ Various forms of Equation 276 can Be utilized to construct a material model. A useful form for materials such as sand can be developed by using Equation 268b and assuming that the ultimate shear strength of the material is a function of J /3 . For a first-order approximation we can assume that the ultimate shear strength is a linear function of hydrostatic stress; thus, 1 (277) k k2 + k 3 3 2 where and are material constants that must be determined ex perimentally. Utilizing Equations 277, 268a, and 268b in Equation 261+ results in the following constitutive relationship 89 al l a. . P 6 . . ij + PB )e B ij aIl 1 k2 * k3 (P0 + PB>e - PB'I (278) aIl k.jk2 + k1k 3 (PQ + PB)e - PB It is noted that when dependence of shear strength on hydrostatic stress disappears (i.e., when k^ = 0) Equation 278 reduces to Equation 269. 76. To examine the significance of the dependency of shear strength on hydrostatic stress let us consider the behavior of Equa tion 278 under cylindrical state of strain (Equation 255)- Eor this state of deformation V f = (e^ - e^)//3 and 1^ = + 2e^ and we can, after arranging terms, obtain the following relationship ~ a2^)klk2 + klk 3 ( " k2 * k 3 (°1 ' °2) + ”2] - ^ (01 - ° 2 1 If we consider a stress path where = a i s kept constant during the test (Figure 6d), Equation 279 can be used to relate stress difference (cr^ - o^) to strain difference (e^ - e^) for a constant value of o^ . The qualitative behavior of Equation 279 is depicted in Figure 10. It is observed from Figure 10 that the shear strength of the material in creases with increasing confining stress . If k^ is set to zero in Equation 279 it is noted that the dependency of shear strength on con fining stress disappears. 77* The next step in developing more complicated quasi-linear elastic stress-strain relationships is to assume that volumetric strain 1^ is caused by both the hydrostatic and deviatoric stresses. For such material we can write (280) 90 ° l - ° 2 CONSTANT Figure 10. Response predicted "by Equation 279 tor cylindrical state of strain where the function g must he postulated, based on experimental re sults, for any material of> interest. Equation 280 can be simplified further by expressing the volumetric strain I as the sum of two com ponents, i.e., h = ®h(r)+ gs(^s) <281) where g = contribution due to hydrostatic stress and g = contribu- n s tion due to deviatoric stresses. For the contribution due to hydro static stress we can use the inverse of Equation 268a + P.B (282) P0 + PB As a first-order approximation, for the contribution due to deviatoric stresses we can express g as s 91 3 - ^ (283) where a is a material constant that can he positive or negative de pending on-whether the material contracts or expands, respectively, dur ing the application of deviatoric stresses. In view of Equations 28l through 283 the relationship for the volumetric strain (Equation 280) becomes r + pB I.. = — £n 1 a (28U) P0 + PB During a hydrostatic test (Figure 6a) = 0 and Equation 28^ reduces to Equation 282. To formulate the constitutive equation for this class of quasi-linear material we combine Equations 268b and 277 to develop an expression for , and then substitute this expression and Equa tion 2Qk in Equation 265* The resulting constitutive equation becomes 1 + P .B e. . 6 3 . . ij P0 + PB ij k.n + (285) k2 + k 3 3 78. Equation 285 allows for the dependency of shear strength on hydrostatic stress and the coupling of volumetric strain and deviatoric stresses. If the material constants k^ and a are set to zero these cross-effects will disappear and Equation 285 will reduce to Equa tion 270. It should be noted that these cross-effects (in particular the coupling of volumetric strain and deviatoric stresses) are due to scalar nonlinearity (invariants) and are different from the second-order effects discussed in the development of constitutive equations for non linear elastic materials. To illustrate this point further, let us examine the behavior of Equation 285 under a simple shearing stress 92 defined by Equation 239* For this state of stress (taking Pq to be __ O zero) <1^/3 = 0 and Jg = x and from Equation 285 it follows that e .. = I = ax (286) 11 1 indicating that simple shearing stress is accompanied by volume change. However, Equation 285 indicates that there are no normal deviatoric strain components associated with the volume change, i.e., E11 E22 = E^^ = 0 . In the case of nonlinear elastic material, on the other hand, it was shown that there are normal deviatoric strains associated with volume change due to simple shearing stress (see Equation 2*4-2). Finally, it should be pointed out that for certain stress paths Equa tions 278 and 285 will produce inelastic stress-strain response (these stress paths will be discussed in Part V in conjunction with incremental stress-strain laws). This is a consequence of the dependency of the shearing response of the material on the hydrostatic state of stress (Equation 2j6). To avoid such possibility the use of Equations 278 and 285 should be restricted to stress paths where J^/3 and do not decrease, i.e., they remain constant or increase. Green’s Method 79. In order to develop the constitutive equations of elastic ma terials based on Green's method we first state two of the fundamental laws of mechanics:"’ a^. The first law of thermodynamics, which is a statement of the law of conservation of energy: "The work that is performed on a mechanical system by external forces plus the heat that flows into the system from the outside equals the increase of kinetic energy plus the increase of internal energy." b. The law of kinetic energy: "The work of all the forces (internal and external) that act on a system equals the increase of kinetic energy of the system." Mathematically, we can express the first law of thermodynamics as'’ ¥ + Q = AT + AU (287) e 93 where We = work performed on the system by external forces Q = heat that flows into the system AT = increase of kinetic energy AU = increase of internal energy The law of kinetic energy can be stated as W + W. = AT (288) where W.^ = the work performed by internal forces in the system. In view of Equations 287 and 288 we can write W. = Q - AU (289) Since we are only dealing with mechanical energy we assume that Q = 0 and Equation 289 reduces to W. = -AU ( ) 1 290 80. We will now proceed to derive the constitutive equation of“ Green’s elastic materials. If V is a material volume (region) within a deformable body and S is the surface enclosing this region, and if this region undergoes an infinitesimal displacement 6ui (the symbol 6 defines a small variation), the work of external forces can be expressed as <5W a. .n.6u. dS + f . <5u. dV (291) e Ji J i 1 l where n = direction cosines of the outward normal to surface S . The J first integral in Equation 291 is due to tractive forces on S and can be transformed to a volume integral by using the Divergence Theorem,^ i.e., ct .. n. 6u. dS = 6u.a dV (292) Ji J i i The second integral in Equation 291 is due to body forces f^ acting on material in V . Combining Equations 292 and 291 5 the work of external forces becomes 5W . (<$u. ) 6u . ( a .. dV (293) ■ / / y - , 1 1 . i Ji. + f i>] Now, let us assume that during the displacement 6u^ the material vol ume V is in equilibrium and the change in kinetic energy is zero. The equations of motion (Equation 192) then take the form a. . . + f . = 0 (29*0 ij ,3 i which are referred to as the equations of equilibrium. Since the stress tensor is symmetrical, application of Equations 29^ in Equation 293 re sults in the following expression for 6¥ g <$W = iff a..(6u.) . dV (295) e JJJ ji i ,j Similar to Equations 1^9 through 151, the infinitesimal displacement gradient (6u.) . can be expressed as 1 5 3 (6u.) . = 6e.. + 5ft.. (296) 1 iJ ij In view of Equation 296, Equation 295 becomes 5W a.. 5ft dV (297) e Ji iJ,) Since a.. = a.. (symmetry of stress tensor) and 6ft.. is skew- Ji ij iJ symmetric, the expression i-s zero an^- Equation 297 reduces to SW a..5e. . dV (298) e iJ ij Invoking the assumption that AT = 0 during the infinitesimal displace ment, and since Q = 0 , Equation 287 can be written as 95 <5We = 6U (299) The internal energy associated with the material volume V can be ex pressed as U = U„ dV (300) where UQ is the internal energy per unit volume, referred to as the internal energy density. In view of Equations 298, 299, and 300, we can write cr. . 6e. . dV 6Uq dV (301) ij ij which leads to su a . .6e . . (302) 0 ij ij Since the internal energy density function U depends on the strain 0 components e. . the variation 6U^ due to 6e. . can be expressed as ij 0 ij 3U„ 0 6 e . . (303) % - 3s. ij ij In view of Equations 302 and 303, the stress tensor a . . takes the fol- ij lowing form 3tL a. . (30U) ij 3e. . ij Equation 3 (A is referred to as the Green elastic constitutive equation. 8l. For isotropic materials the strain energy function UQ must be invariant and, thus, a function of strain invariants. Therefore, for isotropic materials we can write 311 . ( v cr. . 305 ij 3e. ( ) ij 96 If the material -under consideration is incompressible, I = 0 and U q = , ^3)' ^or suc-*1 material, an arbitrary hydrostatic state of stress may be superimposed on the existing state of stress given by Equation 304. Using chain rule of differentiation, Equation 305 can be expressed as 9U0 9 I2 9Uo 9Ii 9U0 9I3 (306) »7 9 e.. °iJ = 9I1 9£ij 3I2 iJ 9I3 9£ij Since 3Ii (307a) 91. (307b) 9eij = "J 31. e . £ . (307c) 3e. . lm mj ij Equation 306 can be written as 3U„ 31L 3LL a. . = 6 . . + e . . + e . e (308) ij 31-l ij 9 ij 3Ï immj Comparison of Equation 308 with Equation 197 indicates that the Green and the Cauchy elastic constitutive equations have the same form. The difference between the two formulations is that the response coeffi cients (j> , 4»^ , and 97 '8U, 8 8 31, (309a) 81, auo\ 8 8 81, (309b) 81 3 '8U„ 8 81, (309c) 81 31, 3 The consequence of the above restrictions imposed on the response coeffi cients -will be realized when we develop second- and higher-order elastic stress-strain laws using Equation 308. The Green elastic material can, therefore, be considered as a special type of Cauchy elastic material where the response coefficients are restricted by Equation 309. 82. The inverse of Equation 308 (the counterpart of Equation 198) can be determined by assuming that there exists a function so that Ur r_ = a..e.. (310) 0 ij ij Equation 310 holds for elastic materials where application of a positive stress increment results in a positive strain increment and vice versa. The function is referred to as the complementary energy density function. From Equation 310 it follows that + a. ,e. . (311) ro = -uo ij ij Differentiating Equation 311 with respect to yields 98 9Ua 9e.. 9a.. ÜI -5-+ O.. + -JJ- (312) 9ct 9a ij 9 a. ij 9a. mn mn mn mn Since U Q is a function of strain it follows that 9U, 9th 9e.. 0 i: (313) 9a 9e. . 9a mn ij mn Combining Equations 313 and 312 results in 9f, 9a. 9U, 9e. iJ = e . . J-J. (3lU) 9a ij 9 a. aij 9e 9a mn mn IJ/ mn In view of Equation 30^ the second expression in Equation 3lU is zero. Equation 31^- then becomes 9f, 9a. . = e 3-J (315) 9a ij 9 a. mn mn Since 0 i ^ m or j # n ! (316) 1 i = m , j = n Equation 315 becomes 9f, e (317) mn 9a mn or e . . (318) ij Equation 318 is the inverse form of Equation 30U. 83. For isotropic materials is a function of stress invari- ants given by Equations 118 through 120, i.e., 99 Using chain rule of differentiation, Equation 319 can he expressed as 3F0 9J1 8r0 8J2 e . . (320) ij 3J, 3a.. 9er- • .t - 9a. . 1 iJ 9J2 iJ 9J3 « Since 3J 1 6. . (321a) 3a. . ij ij 3J 2 a. . (321b) 3a. . ij ij a. a . im mj (321c) Equation 320 takes the following form 3Ih e . . ô. . + + a. a . (322) ij 1 3 un mj Equation 322 has the same form as Equation 198. The response coeffi cients in Equation 322, however, are not independent. It can readily be shown that relations similar to Equation 309 exist between these coefficients. 84. The complementary energy density function can also be expressed in terms of strain invariants by utilizing Equations 308 and 310. From Equations 308 and 310 it follows that r„ = a . . e.. - U. ij iJ 0 '3U 3U 3U ^JT- 6. . + — - e. . + — - e. e . |e. . - U, 3 ^ 10 io 3y im 9Uq _ 3U _ 3U (323) ■I^ +2I* ^ t3I^ - Do Since for a given material is known, we can use Equation 323 to ex press the complementary energy density function in terms of strain in variants. However, to obtain an inverse constitutive relationship (Equation 322) we need to express rQ in terms of stress invariants. This can be accomplished, at least in principle, by first expressing the stress invariants in terms of strain invariants using Equation 308. The resulting expressions can be inverted to obtain strain invariants in terms of stress invariants and then substituted in Equation 323 in order to express in terms of 5 and <1^ * now develop constitutive equations for various classes of elastic materials utiliz ing Green1s method (i.e., Equations 308 and 322). Linear elastic material 85. For linear elastic materials only terms in strain up to the first power are retained in the stress—strain relationship. It then follows from Equation 308 that the strain energy density function UQ must be quadratic in strain in order for the resulting stress-strain re lation to be of first order. Assuming that the initial state of stress is zero, U q takes the following form IF (324) A1I2 + A2I1 where A and are material constants. Substituting Equation 324- in Equation 308 results in a . . = 2A0I 101 Equation 325 is identical with Equation 201, indicating that both Green's method and Cauchy's method result in the same stress-strain rela tion for linear elastic materials. In view of Equations 20^ and 209 the material constants A2 and A become ^2 2 — 3 (326a) Ax = 2G (326b) The strain energy density function for linear elastic materials (Equa tion 32*0 can then be expressed in terms of shear modulus G and bulk modulus K as follows : uo - 20 (h - \ 4 ) + I X1 (327) It is noted that the expression ^ 2 - is secon(^ invariant of strain deviation tensor 1^ (Equation 16^4-). We can therefore write U0 =2Gl2 + l I? (328) Since for linear elastic materials J^/3' = KI^ and = 2G V n (Equa tion 266), the strain energy density function can also be written as u0 = + (329) where V -^2 an<^ Gl ^ i ^ can considered as energy due to distor tion and volume change, respectively, during a deformation process. The strain energy density function can also be expressed in terms of stress invariants or various other combinations of elastic moduli and invari ants. For example, the counterpart of Equation 328 becomes 2 ~ (330) 102 In view of Equations 323 and 327 the complementary and strain energy density functions for linear elastic materials are identical. For linear elastic materials we can, therefore, write (3 3 1 ) U 0 r0 2 aijGij 86. We can now examine the nature of the restrictions that must he placed on the linear elastic moduli due to the existence of strain energy density function . Expanding Equation 32U, we can write (332) + 2A2 (ell€:22 + e22e33 + £lle33) Equation 332 is quadratic in strain and can be expressed in the qua dratic form 6 6 U . . e. e. (333) 0 E Eeij i J i=l j=l where e and e. denote the six independent components of the strain i 3 tensor. The matrix c . . = c .. is expressed in terms of A and ij A2 and has the form 0 0 0 >♦ A2 A2 A2 0 0 0 A2 A2 A2 0 0 0 c . . = A2 A2 A2 (33U) ij 0 0 0 0 0 Ai 0 0 0 0 0 A1 0 0 0 0 0 Ai 103 The strain energy density function UQ is a positive quantity. Accord ing to the theory of quadratics, in order for Un to he a positive quan- tity all the minors of the diagonal elements of c.. must he positive. Imposing this restriction on the diagonal miners of the matrix c.. ij leads to the following inequalities: V > O (335a) H A_ (335b) I T + A 2 > 0 A 1 + 3A2 > 0 2 (335c) In view of Equations 326, 219, and 220, the above inequalities impose the following restrictions on the linear elastic moduli G , K , E , and v : G > 0 (336a) K > 0 ( 336b ) E > 0 (336c) -1 < V < I 336 2 ( d) It is emphasized that negative values of Poisson’s ratio v have not been found experimentally for isotropic elastic materials. Nonlinear elastic material 87. We will now proceed to develop constitutive equations for various classes of nonlinear elastic materials within the framework of Green's method (Equations 308 and 322). As in the development of non linear Cauchy elastic constitutive equations, we will start with second- order stress-strain relation where terms up to the second power are re tained in the stress-strain relationship. The strain energy density function for a second-order stress-strain law, therefore, must be cubic 10l* in strain. Assuming that the initial state of stress is zero, the most general cubic relation for takes the following form U0 = 2GI2 ♦ (337) (!-§ ) + V l L + V l + A 5I3 where through are additional material constants associated with second power terms in the stress-strain relationship. It is noted that if third power terms in strain are neglected Equation 337 degen erates to the corresponding expression for linear elastic material (see Equations 32k and 326). Substituting Equation 337 in Equation 308 we obtain the following second-order stress-strain law a. . 6. . ij + 3Vl + V2 ij + (2G + A_I, )e. . + A._e. e . (338) 3 1 ij 5 îm mj Equation 338 contains five material constants whereas its counterpart based on Cauchy's method (Equation 2^8) contains six. This reduction in material constants is a consequence of thermodynamic restrictions im posed on the response coefficients of Green elastic material (see Equa tion 309). It was noted that in the case of linear elastic materials both methods resulted in the same equation and their difference was not apparent. The difference between these two methods becomes more pro nounced when considering higher-order nonlinear elastic and quasi-linear elastic materials. 88. Let us now consider the stress-strain relationship for third- order elastic materials. The strain energy density function for a third-order elastic material must contain strain terms up to the fourth power. Again assuming that the initial state of stress is zero, the strain energy density function for a third-order elastic material takes the following representation U0 - 2GI2 ♦ (f-f) A + VlL + Vl + Vs + VÏ (339) + A-j T2 + A8I1I3 + A9I1I2 105 where Ag through are additional material constants. It is noted that when fourth power terms in strain are neglected Equation 339 re duces to Equation 337 (strain energy function for second-order material) Substituting Equation 339 in Equation 308, we obtain the following third order stress-strain law a. . 6. . ij h + » A 1! + A3X2 + V l + V 3 + ij 20 + + 2A I2 e . e . (3^0) + "'S1!)"« + (A5 + A8ri> un mj Equation 3^+0 contains nine material constants. The counterpart of Equa tion 3^0 based on Cauchy's method (Equation 259), on the other hand, contains 12 material constants. Therefore, in the case of third-order stress-strain law the effect of thermodynamic restrictions is to reduce the number of material constants by three. In the next section we will consider quasi-linear elastic materials within the framework of Green’s method. Quasi-linear elastic materials 89. If the strain energy density function is independent of the third strain invariant I^ , Equation 308 reduces to cr. . e . . (3kl) ij ij sq sij 31. Equation 3^1 is the counterpart of Equation 26l (constitutive equation of quasi-linear elastic materials based on Cauchy’s method). In the case of Equation 3^1, however, the response coefficients are restricted by Equation 309a. From Equation 3^1 it follows that fV 1 T !üo (3^2a) 9I1 3 1 Æ . s.. — ^ E. . (3^2b ) IJ 3I2 1J 106 Using Equations 124, 162, and 3^-2, we can obtain expressions similar to Equation 263 for the response coefficients in Equation 3^1, i.e., 9U 0 (3^3a) 91 2 9U 0 (3U3b) 9Ï 1 The inverse of Equation 34l, leading to strain-stress relationship, takes the form 911 e . . fia, + (344) ij 3J, 5lj 90. In order to examine the nature of the restriction placed on the response coefficients by Equation 309a, we substitute Equation 343 in Equation 309a as follows : (345) Therefore, the functional forms of J must satisfy the above differential equation. In the case of quasi-linear elastic mate rial based on Cauchy’s method, it was noted that the functional forms of and were not restricted. If we consider a material for 1 which depends on I only (i.e., volumetric strain is caused only by hydrostatic stress), then Equation 3^5 reduces to FI (3U6) 3 3l2 \ > | T 2. N 107 Equation 3k6 can be satisfied only if V ^ 2 / ^2 ;'‘s ^orm = f2 ( i p (3Vr) Equation 3^7 indicates that for a quasi-linear elastic material for which depends only on I , the shearing response of the material is independent of hydrostatic stress. In view of Equations 3Ul, 3^3, and 3^7, the constitutive equation for this class of material becomes " U ‘ + [f2<^>] («ij - I V i j ) <*8) The functions f and f^ can he postulated, based on experimental evidence, .for a given material. For example, Equation 269 is a special case of Equation 3^8 where the functions f and f^ are obtained from Equations 268a and 268b, respectively. 91. Equation 3^8 is the simplest form of equation for quasi- linear elastic material in that there is no coupling between the devia- toric and the volumetric responses of the material. Equations for more complicated forms of quasi-linear elastic materials can be developed by expressing the strain energy density function UQ as a polynomial func tion of I_j and Ip , or by postulating mathematical expressions for Ip and J^/3 that will satisfy Equation 3^5 and will include various degrees of coupling as desired. 108 PART V: INCREMENTAL CONSTITUTIVE EQUATIONS 92. Incremental constitutive equations are often used to describe the stress-strain behavior of materials in -which the state of stress is a function of the current state of strain as well as of the stress path followed to reach that state. The general form of the constitutive equation for this class of material behavior is generally expressed as a. . F. . (e a ) (3 U9 ) ij ij mn rs where F. . is a response function. Equation 3^-95 which is a special J case of Equation 19^, expresses the components of one tensor in terms of the components of two other tensors. Therefore, in view of Equation llU, the functional form of F . . takes the following representation cr. . + nn e . . + n_e.. e, . + rua. . ij V i j 1 ij 2 lk kj 3 ij + n.a.-i o. . + . + a., ê, .) U ik kj 5 lk kj ik k j ' + iv(è.., e, o . + a., ê, e . ) 6 ik kp pj ik kp pj + ru ie ... a, a . + a., a, e . ) 7 ik kp pj ik kp pj lQ( è è , a , a, . + a., a, ê .è, . ) (350) o ik kp pt tj ik kp pt tj where the response coefficients r)Q , .. . , rig are polynomial func tions of the invariants of £ and a and the following joint mn rs ° invariants n (351a) 1 e ah xO-u ha n (351b) 2 eah°hcaca 109 n = i i a (351c) 3 alD Idc ca n, = ê _ e, a ,a, (351d) 4 ab be cd da 93. We can simplify Equation 350 by assuming that the materials of interest are rate independent. To eliminate time effects, Equa tion 350 must become homogeneous in time. This can be accomplished by eliminating all terms containing second and higher powers of ê in Equation 350. Accordingly, the response coefficients Pg , rig , and n2 must vanish, p^ , P^ , and p 1 must be independent of and functions of stress invariant alone, and p^ , Pg , and Pq must be of degree one in . Imposing the above restrictions on the response coefficients in Equation 350, we obtain «ij ' "oSij + V iJ + V iJ + Vik°kj + + "ikV + p ^ e .-cl o . + a. a è .) (352) 7 ik kp pj lk kp pj The response coefficients p^ , Pg , and Pq can now be written as (353a) n0 ^O1! + gini + P2n2 P3 = e3i1 + eun1 + e5n2 (353b) (353c) % = e6Tl + e7ni + e8n2 where, similar to p^ , P^ , and r\± , the response coefficients $ ... , gg are independent of and functions of stress invari ants alone. Substituting Equation 353 in Equation 352, we obtain the following incremental constitutive equation for rate-independent materials 110 (35*0 Equation 35^+ is the most general incremental constitutive relationship for rate-independent material. It contains twelve response coefficients which are polynomial functions of stress invariants. Since each term in Equation 35^ contains a time derivative d/dt (i.e., Equation 35^ is homogeneous in time), both sides of the equation can be multiplied by dt , resulting in the following differential form ab■u °-u ba + (B_ de + 3j, de. a + 3q de 3 nn 4 ab ba 5 (355) where de and do.. are referred to as the strain increment and ij iJ stress increment tensors, respectively. From Equation 355 it is ap parent that incremental constitutive equations are first-order differen tial equations. To obtain unique solutions to these equations we need to prescribe some initial conditions. The integration of the differen tial equations for a given stress path and initial condition will lead to stress-strain relationships. Various classes of incremental con stitutive equations can be obtained from Equation 355 by specifying the highest degree of stress that appears on the right side of the equation. In the following sections we will develop and examine various classes of incremental constitutive equations. Incremental Constitutive Equation of Grade-Zero 9^. If the right side of Equation 355 is independent of stress, the incremental constitutive relationship is referred to as grade-zero. In this case r'T = ^ = &2 = 33 = 0^ = ^ = 3g = 0 , and n1 and become constants. Thus, for grade-zero, Equation 355 reduces to da = g de ó. . + tu d e . . (356) ij 0 nn ìj 1 ij It is noted that Equation 356 has the same form as the constitutive equation of linear elastic material (Equation 201). In order to in clude the linear elastic stress-strain law as a special case of Equa tion 355, the material constants (0, 0, 0) and (0, 0, 0) will be replaced by K - — G and 2G , respectively. Accordingly, Equa tion 356 may be expressed as da. . de 6.. + 2G de.. (357) ij H nn ij ij Equation 357 is the constitutive equation of linear elastic material (see Equation 210) expressed in incremental form. In order to obtain a relation between stress and strain, Equation 357 must be integrated. If we adopt the condition that a.. = 0 when e . . = 0 , integration of J -L J Equation 357 results in the same stress-strain relations as predicted by linear elastic constitutive equation. For example, consider the condition of uniaxial strain (Equation 221) where = de1 . For this state of deformation Equation 357 results in 112 da + 2G de± (358a) 1 da (358b) 2 Integrating Equation 358 and using the condition that an = a_ = 0 when = 0 , ¥e obtain + 3 Gje1 (359a) -I Gk (359b) which are identical with the relationships predicted by linear elastic stress-strain law (Equation 222). Equation 357 > which is the simplest form of incremental constitutive relationship, therefore does not mani fest any nonlinear behavior. Incremental Constitutive Equation of Grade-One 95* If terms up to the first power of stress are retained in the right side of Equation 355 > we obtain the incremental constitutive equa tion of grade-one. This can be achieved by allowing the response co efficients ri^ , Bg , $rj , Bg , B^ , B^ , and B2 to vanish, and the remaining response coefficients to take the following forms eo = K - | g + x 1j 1 (3 6 0 a ) (360b ) Bi = h (3 6 0 c ) *3 = l 3 (360d ) \ = 2G + V i (3 6 0 e) n 5 = *5 113 where A^ through A^ , K , and G are material constants that must he determined experimentally. In view of* Equation 360, the incremental constitutive equation of grade-one becomes de 6. . + A0 de , cr 6 . . aoid =(K-fG +Vi) nn lj 2 ah ha i j + x3 aenn°lj + (2G + V l ) deij + V dEik % + dik d£kj > (36l> Equation 36l contains seven material constants. It is noted that Equa tion 36l reduces to incremental constitutive equation of grade-zero (Equation 357) when the material constants A^ , A^ , A3 , A^ , and A^ vanish. For a given initial condition and stress path, Equation 361 can, in principle, he integrated in order to obtain stress-strain rela tionships. The differential equations generated from Equation 361 for various states of stress and deformation are coupled first-order equa tions. It is not possible, in general, to obtain closed—form solutions for these equations. Therefore, numerical integration schemes must he u t i l i z e d . 96. Let us examine the behavior of Equation 361 under hydrostatic state of stress (Figure 6a). For hydrostatic state of stress, Equa tion 361 results in 40ii - dJl ' 3 (K - f G + V i ) “ l + V l dIl + X3JX dlx + (2G + dlx + I X d l1 (362) Equation 362 can he written as d Jn d Ii = ( 363) |s k + ^ 3Xi + X2 + X3 + H + f *5H] Integrating Equation 363, we obtain the following relationship between volumetric strain and pressure J1 dJ1 / 3K + K J (36Ua) 0 0 or (36UTd ) where K_ = 3An + + A, + 2Xc/3 . We can invert Equation 3 6 k b 112 3 4 ? to relate pressure to volumetric strain, i.e., J 1 3 (365) It is of interest to note that Equation 365 has the same form as Equa tion 268a 9 which was postulated to describe the pressure-volumetric strain behavior of quasi-linear elastic materials. In fact, it is noted that the ratio K/K_ in Equation 365 corresponds to P«, (the maximum tensile stress that the material can sustain before it breaks) in Equation 268a. It is important, however, to realize that unlike Equa tion 268a, Equation 365 is the outcome of a theory (i.e., the incre mental constitutive equation of grade-one). 97* In order to examine the coupling of the deviatoric and volumetric responses of the incremental constitutive equation of grade- one we will examine the behavior of Equation 361 for a state of simple shearing deformation. The strain increment tensor associated with simple shearing deformation is given as 0 0 d£12 0 0 21 (366) 0 0 0 115 where it follows that de = 0 . For this state of deformation, Equa- nn tion 36l results in the following expressions for the nonvanishing components of stress increment tensor (367a) (367b) Equation 367 is a set of first-order differential equations which must he integrated to relate stresses to shearing strain • Without going through the process of integration, however, we can draw certain conclusions about the response of the material in simple shearing de formation. First, to maintain a simple shearing deformation (no volume change), normal stresses must he applied to the boundaries of the speci men. Second, since two of the normal stresses are unequal, Equation 367 predicts the occurrence of normal deviatoric stresses on the shearing planes. It is recalled that the same behavior was predicted for non linear elastic material (Equation 235)* As was pointed out previously, the occurrence of normal deviatoric stresses on the shearing planes is a second-order effect due to tensorial nonlinearity (in this case the last term in Equation 361) and is a departure from linear theories where such nonlinearity does not exist. Incremental Constitutive Equation of Grade-Two 98. If we retain terms up to the second power of stress in the right side of Equation 355, the resulting incremental constitutive equa tion is referred to as grade-two. Thus, for incremental constitutive equation of grade-two the response coefficients 3g , 3^ , and vanish, and the remaining response coefficients in Equation 355 take the following forms (368a) 116 (368b) 31 = X2 + X8J1 (368c) e2 ” X9 (368d) g3 = X3 + A10J1 (368e) " X11 (368f) g6 " X12 = 2G + x ^ + x13j1 + xiUj2 (368g) (368h) n5 = X5 + X15J1 (368i) 117 " X16 where X through A^ c , K , and G are material constants. In view ‘l ... l6 of Equation 368, the incrementali constitutive equation of grade- ■two takes the following representation G + An J-, + A^J^ + | t da ij ” (K - 3 u "r Al°l "H A6°l " ,lTu2 f nn ij + (Xg + deab aba6ij + X9 d£ab abcaca6ij + (X3 + ^10J1 ) denn ai j + h i deab ab a °ij + X. „ de a .. a. . 12 nn lk kj + (2G + X ^ + X13Ji + h v V deij + (X5 + X15Jl )(d£ik °kj + aik dekJ} + X_^(de._ a. a . + a., a, de ) (369) 16 lk kp pj ik kp pj Equation 369 contains l8 material constants. It is noted that Equa tion 369 reduces to Equation 36l (constitutive equation of grade-one) when Xg through, are set to zero. Similarly, we can develop in cremental constitutive equations of grade-three (or higher grades) by retaining the third power (or higher powers) of stress in the right side of Equation 355. However, due to mathematical complexities of incre mental constitutive equations of grade-one or higher, they are seldom utilized to solve actual engineering problems. For this reason, a class of incremental constitutive equations, usually referred to as variable- T 8 moduli constitutive models, is often used for solution of many en gineering problems. The variable-moduli models are relatively simple in that they do not contain second-order effects due to tensorial nonlinearity. Variable-Moduli Constitutive Models 99. The basic constitutive relation of the variable-moduli models is given by da. . = K de 6. . + 2G(de. . - \ d e ^ 6 ) (370) ij nn ij \ ij 3 nn ij f It is noted that Equation 370 has the same form as the incremental con stitutive equation of grade-zero (Equation 357)* In the case of variable—moduli models, however, the equivalent bulk and shear moduli, K and G , respectively, are assumed to be functions of stress invari ants. Depending on the functional forms of these moduli, various classes of variable-moduli models can be constructed. Since Equa tion 370 does not include any second-order term, or terms involving joint invariants of stress and strain increment tensors, it can readily be integrated (for a given stress path and initial condition) to yield stress-strain relationships. Various classes of variable-moduli models are examined in the following sections. Constant-shear-modulus model 100. As implied, for a constant-shear-modulus model, G is 118 constant and K is a function of stress invariants. If we further assume that volumetric strains are caused only hy changes in pressure (i.e., there is no coupling between volumetric strain and deviatoric stresses), the bulk modulus K becomes a function of J /3 only. As a first-order approximation, let us assume that K is linearly related to pressure, i.e., - K = KQ + ~ (371) where (an initial hulk modulus) and are material constants. Substituting Equation 371 in Equation 370, we obtain the following in cremental constitutive equation for a constant-G model Equation 372 is also a special case of incremental constitutive equation of grade-one (Equation. 361). That is, if we set the material constants X , X^ , X^ , and X^ to zero, and replace K and X ^ with KQ and K^/3 , respectively, Equation 36l reduces to Equation 372. 101. We will now proceed to examine the behavior of Equation 372 under various states of stress and deformation. Under hydrostatic state of stress, Equation 372 reduces to dJ\ (373) dIi = 3K0 + Integration of Equation 373 (with the initial conditions that 1^ = 0 when J = 0) results in the following relationship between pressure and volumetric strain K- 4 (37U) KL 119 Equation 37^ is of the same form as Equation 365 (pressure-'volumetric strain relation for incremental constitutive equation of grade-one). 102. For simple shearing deformation (Equation 366), Equation 372 predicts the same behavior as for linear elastic material (Equation 203). Under conditions of uniaxial strain, Equation 372 results in the follow ing expressions (375) do (376) 3 Since there is no coupling between volumetric strain and deviatoric stresses, we can eliminate J^/3 from Equations 375 and 376 by using Equation 37^. Substituting Equation 37^ in Equation 375 and noting that 1^ = for uniaxial strain condition, we obtain Ki£i da K0e d£l den 1 (377) Equation 377 can now be integrated to relate vertical stress to vertical strain . Using the initial conditions that = 0 when = 0 , integration of Equation 377 results in 4 . + K0 K1E1 . (578) "i = 3 C£i + r r K1 Similarly, we can obtain an expression for the radial stress a 2 K„ a 2 (e^ (379) Kn Combining Equations 379» 378, and 37^, we obtain the following relation for the stress path associated with the condition of uniaxial strain 120 k o + k i T 2G in 1 J2 = K, (380) 0 Equations 378 through 380 indicate that a constant-shear-modulus model, with a hulk modulus which is linearly dependent on pressure (Equa tion 3 7 1 )5 can predict nonlinear stress-strain relationships under uni axial strain condition. 103. We will next examine the behavior of Equation 372 under conditions of uniaxial stress. For uniaxial state of stress, = a^ and Equation 372 results in the following expression for the increment of axial stress da. da (k . + K, de + 2G /de. - \ de (381) \ 0 1 3 j nn 1 1 3 nn The increment of volumetric strain de can be eliminated from Equa- nn tion 381 by using Equation 373 (this can be done because volumetric strain is a function of pressure alone); thus, da. 2Gda, da1 = + 2G de1 - (382) 3(3KQ + K La1 ) Integrating Equation 382 (with initial conditions that = 0 when e1 = 0) we obtain 3K0 + Klgl in (383) E1 = 3G 3K 3Kn Equation 383 relates axial strain e^ to axial stress Similarly, we can obtain an expression for radial strain in terms of e2 = e3 axial stress 3Ko + K1°1 1 in (38M £2 = 3K, 3Kn 121 Again it is noted that highly nonlinear stress—strain curves can result from a constant-shear-modulus constitutive model. lOU. Since the shear modulus is constant, Equation 372 can he integrated to yield a total stress-strain relation similar to that for quasi-linear elastic models. In terms of the stress and strain devia tion tensors, Equation 372 results in (assuming zero initial conditions for both stresses and strains) S.. = 2GE.. (385 ij ij In view of the definitions of S.. and E.. (Equations 122 and l6l, -LJ ij respectively), Equation 385 can be written in terms of the stress and strain tensors It is noted that Equation 386 is a special case of quasi-linear elastic material (Equation 261+). Substituting Equation 37*+, for J /3 , into Equation 386, we obtain the following quasi-linear stress-strain rela tionship for the constant-shear-modulus model (387) It is noted that Equation 387 satisfies Equation 31+6, indicating the existence of a strain energy function for the constant-shear-modulus model. Using Equations 3l+3 and 387, we can derive the following expres sion for the strain energy function ( 388) It can easily be verified that if we substitute Equation 388 into Equa tion 3I+I we obtain Equation 387. 122 Constant-Poisson*s-ratio model 105. Another type of variable moduli model is the constant- Poisson's-ratio model where, by analogy to linear elastic materials, it is assumed that the ratio G/K is constant. In terms of elastic Poisson's ratio v , this ratio is given by (see Equation 220) G - 3 (1 - 2v) _ ~ (389) K 2 '(i + v')' P From Equation 389 it is obvious that K and G will have similar func tional forms. In order to examine the consequence of this condition, let us assume that the functional form of K is given by Equation 371. As was pointed out previously, Equation 371 indicates that volumetric strains are caused only by changes in pressure. In view of Equa tions 389 and 371 the constitutive equation takes the following form ■It can also be shown that Equation 390 is a special case of incremental constitutive equation of grade-one (Equation 361). 106. The behavior of Equation 390 under hydrostatic state of stress is identical with that of Equation 372. However, the behavior of Equation 390 under deviatoric state of stress is quite different than that of Equation 372. For example, consider a series of constant- pressure shear tests (see Figure 6c) conducted at , Pg , and P^ s where P^ > Pg > • The deviatoric response of Equation 390 for these constant-pressure tests is depicted in Figure 11a. It is observed from Figure 11a that the deviatoric response is dependent on the super imposed hydrostatic state of stress (in this case G varies linearly with pressure). In the case of a constant-shear-modulus model (Equa tion 372), on the other hand, the deviatoric response does not depend on the superimposed hydrostatic stress. Let us now consider two 123 V* STRESS PATH Figure 11. Behavior of constant-Poisson1s-ratio variable-moduli model during constant-pressure shear test combined constant-pressure shear and hydrostatic tests defined by stress paths abed and abe'd1 shown in Figure lib. The deviatoric responses of Equation 390 for these two tests are also depicted in Figure lib. For the stress path abed it is observed that the stress-strain curve during unloading (line cd) is above the loading curve (line ab). This type of behavior results in an energy-generating loop which is contrary to the 12U observed behavior of real material. For the stress path abc'd* the stress-strain relation during unloading is along path c’df, which re sults in permanent deformation. Furthermore, if the stress cycle abcTd?a is repeated several times, the stress-strain response will result in an unrealistically excessive amount of deformation as shown by the dashed lines in Figure lib. In order to avoid these undesirable behaviors, the use of variable-moduli models for which G is a function of hydrostatic stress should be restricted to stress paths where J /3 and remain constant or increase. Variable-moduli models have been used to simulate the hysteretic behavior of earth materials during Q cyclic or near-cyclic conditions. For this type of problem, two sets of expressions are usually specified for the moduli K and G : one set for loading and one set for unloading. A set of criteria or logics are also specified to determine whether the material under considera tion is being loaded or unloaded so that the proper set of moduli can be used. Application of variable-moduli models for treating hysteretic effects will be discussed later. 107. Let us next consider the behavior of a constant-Poisson1s- ratio model under conditions of uniaxial strain. For this state of deformation, Equation 390 results in (391) 1 We can eliminate J^/3 from Equations 391 and 392 by using Equa tion 37^. Substituting Equation 37^ in Equations 391 and 392 and noting that I = for condition of uniaxial strain, we obtain (393) 125 da, de 2 1 (39*0 Integrating Equations 393 and 39^ and using the initial conditions that a^ = = 0 when = 0 results in (395) (396) Equations 395 9 396, and 37^+ can he combined to obtain an expression for the stress path associated with the condition of uniaxial strain, i. e., (397) As anticipated, the uniaxial strain stress path for a constant-Poisson1s- ratio model is linear (see Equations 226 and 227). The constant-shear- modulus and constant-Poisson1s-ratio models are elementary versions of variable-moduli models. More complicated forms of variable-moduli models are discussed below. These models are referred to as nonlinear variable-moduli models in that both the shearing and the volumetric responses are nonlinear and are represented by different functional forms. Nonlinear variable-moduli models 108. More complicated, and perhaps physically more realistic, forms of variable-moduli models can be formulated by expressing K and 126 G as separate polynomial functions of stress invariants. If we make the assumption that there is no coupling between volumetric strain and deviatoric stresses, then bulk modulus K becomes a function of J /3 alone. For a first-order approximation we will adopt Equation 371 for the bulk modulus. A different functional relation can be postulated for the shear modulus G . As a first-order approximation, we will assume that G is linearly related to the first and second invariants of stress tensor, i.e., G = G q + G 1 — + G2J 2 (398) where GQ (an initial shear modulus), G^ , and G^ are material con stants. In view of Equations 398 and 371 the constitutive relationship for this type of variable-moduli model becomes + K. de 6 . . 1 3 >nn ij + 2 de 6 . . (399) (■G0 + G1 3 + G2J2 ,d£ij - 3 nn ij It is noted that Equation 399 reduces to Equation 372 (constitutive equation of constant-shear-modulus model) if G^ and G2 are set to zero. Also, it can readily be shown that Equation 399 is a special case of incremental constitutive equation of grade-two (Equation 369)* 109. The behavior of Equation 399 under hydrostatic state of stress is identical with the behavior of constant-shear-modulus and constant-Poisson’s-ratio models, since the functional form of bulk modulus is the same for all these models. For deviatoric states of stress, however, the behavior of Equation 399 differs considerably from that of constant-shear-modulus and constant-Poisson1s-ratio models. For example, consider a state of simple shearing deformation defined by 127 Equation 366. For this state of deformation, Equation 399 results in (assuming zero initial state of stress and deformation) (UOQa) a°ll = d°22 ' d°33 = 0 da (UOOb) 12 2 (G0 + G2°12) del2 Integrating Equation UOQb we obtain e 12 (hoi) It is noted from Equations 372 and 390 that for this state of deformation the constant-shear-modulus and constant-Poisson1s-ratio models predict a linear relationship between the shearing strain and the shearing stress °y 2. # ^ should also be noted that within the framework of variable-moduli models normal stresses are not required in order to maintain a state of simple shearing deformation. As was shown pre viously, to maintain a simple shearing deformation in the case of in cremental constitutive equations of grade-one (or higher grades), normal stresses must be applied to the boundaries of the specimen (see Equa tion 367 ). 110. A more useful description for shear modulus, especially for 8 . modeling the stress-strain behavior of soil, is to express G in terms of Jx/3 and rather than • For example, for a first-order approximation we can express G as (U02) G = G0 + G1 - + °21 where G^ is a material constant. The sign of G^ will determine whether the material softens (G^ < 0) or stiffens (G^ > 0) during shear. For this description of shear modulus the constitutive equation becomes 128 + 2 + Gi r (U03) For a state of simple shearing deformation (Equation 366), Equation b03 results in (assuming zero initial state of stress and deformation) do11 = da22 = da33 = 0 (UoUa) (UoUb) 4012 = 2 (G0 + Integrating Equation k0k\) we obtain G0 + G2°12 e12 (!*05) Equation U05 also describes a nonlinear relation between the shearing strain e12 and the shearing stress a^2 . 111. Let us next consider the behavior of Equation U03 under cylindrical state of strain utilizing the stress path depicted in Figure 6d. For this stress path = /3 («q/3 - a ) , where o3 = o2 is the confining stress which remains constant during the deformation process. From Equation 1+03 the increment of stress difference associ ated with cylindrical state of strain becomes (Uo6) d We can eliminate J /3 from Equation Uo6 by using the equation for the stress path, and noting that = (c^ - o2)/J3 , we obtain dai ” da2 de de. 1 (U07) (a, Gia2 + For a given value of confining stress , Equation 1*07 can be inte grated to yield a relation between strain difference e^ - e^ and stress difference ° ± ~ ° 2 * Denoting the confining stress by and carrying on the intégration, we obtain 1 e 1 G0 + G1ac + (a1 - a£) x £n ( 1*0 8 ) G_ + G.a 0 1 c It is noted from Equation koQ that the shearing response of the material is dependent on the magnitude of the confining stress a^ . If is set to zero in Equation Uo8, this dependency disappears (i.e., shear modulus becomes independent of hydrostatic stress, see Equation U02). 112, Next we will consider the behavior of Equation U03 under conditions of uniaxial strain. For this state of deformation, de = dGl and E^ ation yields the following expression for the increment of vertical stress da^ : da 1 (U09) We can eliminate J /3 and from Equation k09 by using Equa tion 37l* and noting that = V3 (a1 - ^ 73)72 ; thus, _dai _ «2S3 3 G2a1 - = K0 + -k G0 + r- K0 K1 + I G1 - ¥ È2, (^10) 1 K1 130 It is noted that Equation UlO reduces to Equation 377 (the corresponding expression for a constant-shear-modulus model) when G^ and G^ are set to zero. Equation UlO is a first-order differential equation and has the following solution (assuming zero initial state of stress and deformation) K0 / ~ It 2/3 — IK, + -k G, - — ■ G, 1 3 1 3 2 , K, e V i - e ¥ V i al = 2/3 ~ h ' - r 13 2 K ° ' h 2 K. + I ° 1 - T 52) - (Ko - 3 Go 2/| ~ 3 G2 è e N 3 T 1 x I 1 - e (^11) Again, it is noted that Equation 1+11 reduces to Equation 378 when G^ and are set to zero. The radial stress required to prevent radial strain can he determined by direct integration of Equation 1*03 (similar to the procedure followed to obtain a^) or by using Equa tions 3 7 and 1+11. From Equation 37^ it follows that (U12) In order to relate a ^ to axial strain we simply eliminate from Equation 1+12 by using Equation 1+11. The uniaxial strain stress path for this model can be determined by using Equations 1+11 and 37^- Equation 37^ can be inverted to relate 1 ^ in this case) to J /3 . The resulting relation can then be substituted into Equation 1+11 to express in terms of J^/3 which, in conjunction with the 131 expression = /3 ( c - J^/3)/2 , will result in the stress path. 113. More complicated forms of nonlinear variable-moduli models can he developed and analyzed as for the preceding models. The choice of any particular model, however, must he based on the experimental observation of the stress-strain behavior of the material of interest. Treatment of hysteretic behavior llh. As was pointed out previously, variable-moduli models have been used to simulate the hysteretic behavior of earth materials during cyclic or near-cyclic loading conditions. To show the procedure by which the hysteretic behavior is simulated, we express the basic consti tutive relation of the variable-moduli models (Equation 370) in terms of the hydrostatic and devidrtoric components, i.e., dJ„ — = K dIi (Ul3a) dS. . = 2G dE. . (Ul3b) ij ij where dS.. and dE.. are, respectively, the deviatoric stress and ij ij strain increment tensors. It is postulated that the basic form of Equation Ul3 is valid for all loading conditions (initial loading, un loading, and reloading). However, the functional forms of K- and G change depending on whether the material under consideration is being loaded or unloaded. As depicted graphically in Figure 12, for initial loadings the response of the material is governed by dJ_ (UlUa) — = K* dIi d S . . = 2G„ d E . . (UlUb) ij & ij where and G^ are, respectively, the bulk and shear moduli associ ated with initial loading. During unloading and reloading we assume that the response of the material is governed by 132 Jl 3 a b. DEVIATORIC RESPONSE Figure 12. Treatment of hysteretic behavior using variable-moduli models 133 dJ, = K dln u 1 (l+15a) dS.. = 2G dE. . (Ul5b) ij u 1J where and are, respectively, the hulk and shear moduli as sociated with unloading and reloading. 115. To complete the specification of the model we need a cri terion to define the loading condition during a deformation process. We adopt as our criterion the quantity dW = a . . de. . (U1 6 ) ij ij which defines the rate at which the stresses do work during the deforma tion process. According to this criterion, dW > 0 defines loading (initial loading or reloading) and dw < 0 defines unloading. The condition dW = 0 is referred to as neutral loading. The neutral states of loading associated with the rate of work criterion impose certain restrictions on the material constants in the constitutive equa tions for loading and unloading and require special considerations. The material constants must he chosen so that the loading and unloading constitutive equations become identical whenever dW = 0 , i.e., neutral loading. This requirement must he met in order to obtain a unique solution for a boundary-value problem involving cyclic loading condi tions. From Equations blk and kl'p it is apparent that variable-moduli models, in general, do not satisfy this requirement. Il6. In view of the definition of deviatoric strain increment tensor we can express Equation bl6 in the following form dW S + 6 dE mn mn mn (Ul7a) or dW S dE mn mn (i+lfb) 13 h We can eliminate dE and dl^ from Equation ^17b by using Equa tion Ul3; thus, S dS J, dJ nTT mn mn . 1 1 dH ' -- 2G-- + ~~9K <1,l8) Since J ’ = 1/2ÌS S V it follows that 2 V mn mn/5 dS mn (Ul9) where dJ^ is the increment of the second invariant of stress devia tion tensor. Equation Ul8 can, therefore, be written as dJ^ dJ1 dW = ~2G + 9K (U20) For variable-moduli models in which G = G(j^) and K = K(j^) the rate of work can he separated into hydrostatic and deviatoric components. Equation U20 can then he used with the interpretation that dJ^/2G is rate of work due to deviatoric stresses and dJ^/9K is rate of work due to hydrostatic stress. Accordingly, two criteria for defining various loading conditions can he prescribed. For the deviatoric part of deformation, loading and unloading are defined according to whether dJg is positive or negative, respectively. For the hydrostatic part of deformation, loading and unloading are defined according to whether dJ^ is positive or negative, respectively. In this manner, it is possible for the material to unload in shear while loading in pressure or vice versa. It should he pointed out that these criteria also do not satisfy the requirement of neutral loading. Because of the require ment of neutral loading, the validity of variable-moduli models for 9 treating hysteretic effects has been questioned. Hysteretic effects and permanent deformation can he treated within the framework of in cremental theory of plasticity without violating the requirement of neutral loading. 135 PART VI: CONSTITUTIVE EQUATIONS OF SIMPLE VISCOELASTIC MATERIALS 117. For viscoelastic materials, the state of stress is a func tion of both the state of strain and time rate of strain. The stress tensor can, therefore, he expressed as a. . = F..(e è ) (tei) ij ij mn rs where F . . = viscoelastic response function. In view of Equation 11^, the response function F.. takes the following form <3 a. . = ru<$. . + rue. . + • + n0è. . ij 0 ij '1 ij 2 lk kj 3 ij + x]) ê._ è, . + ri-fe... L . + è e 4 lk kj 5\ ik kj lk kj/ + x\r (s., e. è . + è., e, e 6 V ik kp pj ik kp pj/ + n7 (Eik^kp^pj + hk^kp£pj) (te2) + n8 (eikekpept£tj + eikekpeptEtj) where the response coefficients t)q , ... , rig are polynomial functions of the invariants of e and e and the following joint invariants mn rs II a e -u£-u (te3a) H ah ha ■(te3b) n2 = £ ah-u£-u he £ ca (U23c) n3 = £ ah-u£-u he £ ca (U23d) nU = GahGhcGcdGda It is noted that when dependence on erg disappears, Equation te2 re duces to the constitutive equation of Cauchy elastic material (Equa tion 197). Various classes of viscoelastic materials can he described 136 by Equation 422 by proper selection of the response coefficients Dq , . . . , rig • We will be dealing with simple viscoelastic materials \ in this report. Kelvin-Voigt Material 118. The constitutive equation of Kelvin-Voigt material, which describes the simplest type of viscoelastic material, can be obtained from Equation 422 by allowing the coefficients r\rj , and rig to vanish and the remaining coefficients to take the fol- lowing forms (424a) "0 = AI1 + V i n1 = 2G (424b) (424c) ^3 = 2Gv Accordingly, the constitutive equation of Kelvin-Voigt material becomes = XI_6. . L- O . . (425) ij 1 ij v 1 IJ IJ The coefficients X and G , analogous to the elastic constants X v v and G , are the dilatational and shear viscosity coefficients, respec tively. Again it is noted that'if X and G^ are set to zero Equa tion 425 reduces to the constitutive equation of linear elastic material. 119. Let us examine the behavior of Kelvin-Voigt material under uniaxial state of strain defined by Equation 221. For this state of deformation, Equation 425 results in a., = As, + X ê, + 2Ge + 2G e, (426a) 1 1 v 1 1 v 1 o0 = Àe + X ê, (426b) 2 1 v 1 where e is the strain in the direction of motion in a uniaxial strain configuration and a2 is the lateral stress required to prevent lateral 137 strain. Equation l*26a can be written in a more compact form by collect ing terms o1 = U + 2G)e1 + U T + 2Gv )e1 (1+27) The term A + 2G is recognized as the constrained modulus M (see Equation 225). Analogous to M , we denote the term A^ by M , which is the viscosity coefficient associated with the conditions of uniaxial strain. Equation 1+27 may now be written as o = Me. + M (1+28) 1 1 v 1 Equation 1+28 is the counterpart of the differential equation of motion for a linear spring and a linear dashpot connected in parallel (Fig ure 13) • 120. We will now examine the behavior of Equation 1+28 for an ap plied constant stress of magnitude . For an applied constant stress c+q , Equation 1+28 becomes = Men + M é, 0 1 v 1 ( 1*2 9 ) Integrating Equation 1+29 (with initial conditions that = 0 at t = 0) we obtain -(M/M )t e e v 1+30 1 ( ) Equation 1*30 describes the strain-time response in uniaxial strain con dition due to an applied constant stress of magnitude (Figure 1 3 ). In view of Equations 1+30 and l+26b5 the lateral stress-time response dur ing application of a^ becomes a 2 (1+31) 121. Let us next examine the behavior of Equation k2Q due to ap plication of a time-varying stress condition defined by 138 o\ e l Figure 13. Behavior of Kelvin-Voigt material in uniaxial strain configuration /_ -Ct\ (U32) °i = ao 1 - e } where c is a constant. Substituting Equation k32 in Equation U28 and integrating the resulting expression (with initial conditions that = 0 at t = O) we obtain cM -(M/M )t 0 -ct 1 + E1 M (M - cM ) M - cM (**33) V 139 Equation 1+33 describes the strain-time response in uniaxial strain condi tion due to application of stress-time history given by Equation 1+32. We can eliminate the time t between Equations 1+32 and 1+33 to obtain the following strain-stress relationship ! i \ + ^ ___ , ! e (U31+) 1 a J M(M - cMy) 1 Equation I+3I+ indicates that the stress-strain response is not unique and is dependent on the constant c . 122. Analogous to nonlinear elastic material, we can construct nonlinear viscoelastic constitutive relationships by retaining some of the second-order terms in Equation 1+22. For example, taking = Hg = p = Pg = 0 in Equation 1+22, a second-order viscoelastic constitutive relationship, often referred to as the nonlinear Kelvin solid, can re sult , i.e. , a. . T)-E. . . . + T)\ e e. . (1+35) ij "o6 ij 1 IJ n 2 Ei k £kj + 03 ij4 4 4 lk kj It is noted that in Equation 1+35 there is no tensorial coupling between the strain and strain-rate tensors. Various classes of nonlinear Kelvin solid can be developed, similar to nonlinear elastic material, by expand ing the response coefficients p , ... , p^ in terms of the invariants of the strain and strain-rate tensors. Maxwell Material 123. Rate-dependent constitutive relationships can also be ex pressed as cr. . = F. . (ê CT ) (1+36) ij ij mn rs Equation 1+36 is identical with Equation 3l+9- The most general form of Equation 1+36 is given by Equation 350 and contains nine response coefficients. If we neglect all second- and higher-order terms in Equa tion 350 (i.e., let r\^ = = r)g = 0), and assume that the remaining coefficients take the following forms t u = -a J_ (i+37a) 0 m 1 ru = 2G (^37b) l m n0 = -23 (l*37c) 3 m & are m; m m m m a.. = (-a J_ + A i. )ô. . + 2G ê.. - 2g a.. (1*38) ij m l m 1 ij m ij m ij Equation 1+38 is the constitutive equation of Maxwell material. It con tains four material constants. It is noted that if we set a and 3 m m to zero and integrate Equation 1+38, we obtain the constitutive equation of linear elastic material (replacing G by G and A by A). m m 12k. Let us consider the behavior of Maxwell material under uni axial state of stress defined by Equation 215. Eor this state of stress, Equation 1+38 results in = (-a J + A I.. ) 2G - 23 a, (l*39a) m 1 m 1 ml ml 0 = (-a J + A In ) 2G ê0 (l*39b) m 1 m 1 m 2 where is axial stress and and e^ are axial and lateral strain-rate components, respectively. Since for uniaxial state of stress = cr^ and + 2e^ 5 Equation 1+39 can be written in the following form or, + (a + 2g )a = (A + 2G )è + 2A i (M O a ) 1 m ml m ml m2 a a = A è., + (2A + 2G )ê_ (MOb) ml ml m m2 We can eliminate ep from Equation Hl+Oa by using Equation ttOb; thus, . . G 33 X + a G + 28 G m + m - , . mm mm mm £1 = 2 I 1 l _ . . __2 / ai ( W l ) ,3G X + 2G 3G A + 2G mm m mm m Denoting (sG^A^ + 2G2^^(A^ + *^m^ ^m (Mastic Young's modulus) and (3GmAm + 2Gm)/(3emAm + amGm + 23mGm) by nm ’ Equation'll can be •written as (kb2) ei e n m m Equation kk2 is the counterpart of the differential equation of motion for a linear spring and a linear dashpot connected in series (Figure ll+). If is suddenly applied and then held constant d = 0 , and integra tion of Equation kh2 results in a steady linear increase of axial strain with time (Figure ll*). 125* Analogous to nonlinear Kelvin solid, we.may develop a non linear Maxwell solid by taking = ng = = ng = 0 in Equation 350. The constitutive equation of nonlinear Maxwell solid then becomes cr. . n. t. . ( ^ 3 ) iJ V i J 1 iJ 2 ik kj 3 ij Equation bk3 can be used to construct various classes of nonlinear Maxwell solid by expanding the response coefficients rig > • • • 5 in terms of the invariants of the stress and strain-rate tensors. Standard-Linear Material 126. Rate-dependent constitutive relationships can also be ex pressed as d. . &. . (a e (kkk) ij uij rs mn Equation kkh reduces to Equation ^36 if dependence on e disappears. FI The simplest form of Equation bhk, which is referred to as the lk2 MAXWELL MATERIAL ° ì n °b t a. APPLIED STRESS Figure lk. Behavior of Maxwell material in uniaxial stress configuration constitutive equation of standard-linear materials, is expressed as d = (— a Jn + A I + A I ^6. . + 2G e.. + 2G e_. . — 23_cf_. (^5) ij \ s 1 si si X , A , G , G , and 3 are material constants. As where ag , s s s s s expected, if G and A are set to zero, Equation ^ 5 takes the form s s of the constitutive equation of Maxwell material (Equation U38 ). 127. Let us consider the deviatoric response of standard-linear material. From Equation 1+^5 it follows that S.. = 2G E.. + 2G E.. - 2ß S.. (UU6) ij s ij s ij s 1J Equation kh6 is the counterpart of the differential equation of motion for a linear spring and a Kelvin-Voigt element connected in series (Fig ure 15). It can he shown that the material constants G , G , and s s 3^ and the parameters of the corresponding spring and dashpot model are related through the following relationships G s 2n s^sq (W7a) JL (UUTb) 23 s G 2p q s s s (1+lffc) If the deviatoric stress (say S ^ ) is SU(3-(3-enly applied and then held constant S^2 = 0 > and integration of Equation UH6 results in the fol- lowing deviatoric strain-time response G \ -(G /G )t Sn30 s S I s s E (UU8) 12 s s/ Equation hk8 is depicted graphically in Figure 15. As indicated in Fig ure 15, the material exhibits an initial elastic response (similar to Maxwell material) E12(t = 0) (UU9) 2G 2<1s and an asymptotic elastic behavior (similar to Kelvin-Voigt material) „ /, x O s S0 (,1s + V (i+50) *12^ ' G 2q p s s STANDARD-UN EAR 2q„ MATERIAL Figure 15 . Behavior of standard-linear material under deviatoric state of stress 128. Similar to the nonlinear constitutive equations of Kelvin and Maxwell materials, we may develop a constitutive equation for a non linear standard solid containing second-order terms in stress, strain, and strain-rate tensors. This can be accomplished by combining Equa tions k35 (nonlinear Kelvin solid) and kk3 (nonlinear Maxwell solid), i.e. , d. . = rj^6. . + q e. . + n0£*i • + . ij 0 ij 1 ij 2 lk kj 3 1 3 * V i A j + V i J + V i k akj (l*51) Various classes of nonlinear standard solid can be developed by expand ing the response coefficients tIq 5 5 Tig in terms of the invariants of the strain, strain-rate, and stress tensors. 129* Constitutive equations of viscoelastic materials can also be formulated in integral forms or series forms with differential oper ators."^ Discussion of these types of constitutive equations is beyond the scope of this report. 1 k6 PART VII: CONSTITUTIVE EQUATIONS OF PLASTICITY 130. Constitutive equations of plasticity are designed to de scribe the stress-strain behavior of hysteretic materials. The basic assumption employed in developing these equations is that for each load ing increment the corresponding strain increment can be considered as being the sum of the plastic (permanent) and elastic (recoverable) strains. Mathematically, the strain increment tensor d e _ is ex pressed as de.. = de. . + de?. (U52) IJ 10 10 where de?. and de?. are, respectively, the elastic and plastic 1J strain increment tensors. The elastic strain increment tensor is given in terms of incremental elastic relation dS. . dJl — iJ- + — — 0 (U53) 2G 9K ij The elastic moduli G and K can be assumed to be constant or func tions of stress invarients, as dictated by test data. However, to be consistent with path dependency of elastic materials, and to eliminate any possibility of energy generation or hysteretic behavior during elastic deformation (see Figure ll), the forms of G and K should be restricted to G = G ( J p ( ^ a ) K = K ^ ) (^5^) During unloading, or during loading where the state of stress is below a specified state referred to as yield stress, the behavior of the mate rial is defined completely by Equation 1+53* At the onset of yielding (when the state of stress is such that the yield stress is reached), and during subsequent loading, the material will experience both elastic and lVf plastic deformation and Equation h ^ 2 will govern the behavior of the material. Therefore, for a complete description of the material we need to specify the form of the plastic strain increment tensor de?. . J Guidelines as to how the plastic strain increment tensor can be spec ified were established by Drucker1"1" by introducing the concept of mate rial stability. For a stable material, the work done by a set of stress increments when applied on a specimen of the material is positive. Furthermore, if the stress increments are removed, the net work per formed by them during the load-unload cycle is zero or positive. If we denote the set of stress increments by da.. , and denote the correspond ing change in the state of strain due to application of da.. by de.. ij iJ then the first condition of stability can be stated as da. . de. . > 0 (1*55) ij ij In view of Equation 1*52, the second condition of stability can be ex pressed as da., de.. - da., de?. = da. . de?. > 0 (1*56) ij ij ij ij ij ij — Equations 1*55 and 1*56 provide guidelines for determining the form of the plastic strain increment tensor. We also need to specify the form of the yield function ^ , which defines the limit of elastic behavior. Depending on the specification of ^ , various types of plastic consti tutive equations can be established. In general, we will be concerned with ideal and work-hardening plastic materials. Ideal Plastic Material 131. For an ideal plastic material the yield function $ (or yield surface) is fixed in the principal stress space, i.e., it does not move or expand during plastic deformation. The yield surface is only a function of stress tensor, or function of invariants of stress tensor for an isotropic material. Unlimited plastic flow takes place when £(o. .) = k , where k is a material constant defining the onset of J yielding. During plastic deformation then à i d a . . = 0 ij (^57) The stability condition for an ideal plastic material is specified by da. . de?. = 0 (1+58) For uniaxial stress configuration, for example, Equation 1+58 indicates that during plastic deformation the stress remains constant while the strain increases. Since all admissible stress increments da.. satisfy- ij ing Equation ^57 must also satisfy the stability postulate given in Equation 1+58, it follows that a< ) = 1 iff: ^59) IJ where A is a positive scalar factor of proportionality and is depen dent on the particular form of the yield function ^ . Equation 1+59 is often referred to as the plastic flow rule. Inherent in Equation 1+59 is the normality condition which indicates that the plastic strain in crement (viewed as vector) is normal to the yield surface ^ . Another consequence of the stability postulate is the convexity condition, which requires that the yield surface ^ must be convex in the principal stress space. In view of Equations 1+53 and'1+5 9 , the complete expression for the strain increment tensor becomes de. . 6. . + A (1+60) ij ij Equation 1+60 prevails in the plastic range (d^ = 0). In the elastic range, and during unloading from a point on the yield surface (djf < 0), the behavior of the material is governed by Equation 1+5 3 . 132. In order to use Equation 1+60 we must determine the form of the proportionality factor A . This can be accomplished by combining Equations ^57 and i+60. From Equation k60 we can determine the stress increment tensor da.. ij dd« = 20 d£ij - 2Gi 3 ^ ♦ (i - 1 ) dJi SI3 e*61’ J Substituting Equation l+6l in Equation 1+57 we obtain 2G de. . - 2GÀ -r^- /l _ 2G) dJ (1+62) da ij 9cr... 9a. . \3 9K/ 1 d¥~ <5. . = 0 ij 10 10 1 3aij 1J We can eliminate dJ^ from Equation 1+62 by using Equation 1+60. From Equation l+60 it follows that dJl = 3K (dIl - S sij) ( W 3 ) In view of Equations k63 and 1+62, the proportionality factor A takes the following form A = (k6k) It is noted that all indices in Equation k6k are dummy indices, indicat ing the scalar character of A . Equations k6k and k60 can now be com bined to give an expression for the strain increment tensor _»i_ ae + 3 K ^ 2 G s dS.. dJ_, 9a mn 6G 1 8a mn -- + --=k £ + mn mn 9 de. . ÌA. _ £_ (W5) ij 2G 9K ij do :. M ___ H - + 3K,„- ,2G (■ * - » Y IJ 3a 8a 6G V 3a mn / mn mn \ mn / J It follows from Equations l+6l, ^63, and b6k that the stress increment tensor takes the following representation 150 _9i_ de + 3K.-„2G dI _il_ ,5 8a .. demn 6g dll 8a V mn mn da. . = 2G dE. . + K d L 6. . - ij ij 1 ij dA 3K - 2G 3a 3a 6G l3aA**-« mn vI mn mn \ mn / + 2G (k66) ^ 3 “ y 10 3aij In order to use Equations h6$ and k66 we only need to define the form of the yield function for a particular material of interest. For a number of engineering materials, particularly soils, the yield func tion is generally expressed in terms of J and Vi■ i.e. , <(oij> = <(J1 ' V ^ ) = k (1>6T) For the above specification of ^ it follows that _ jlji____1_ . 2 = M- + _1 ____ 9 1 S. . ( 1+68) 8a.. 8Jn 8a.. ij 1 ij 8aij 9J1 lJ 2 ^ 8>/jJ Application of Equation 1+68 in Equations 1+65 and 1+66 results in 3K ai |f- + - ^ = -*■&— S dE d S .. dJ 1 3Ji VjJ aVF ™ d£ij = + 9 T 6ij + >/ff a.. + (1+69) 9Ji 1J 2 \ j i 8\ j I 1Jy and 151 dE mn 2G dE. . + K dl ô. . IJ 1 iJ (UTO) Equations 1+69 and 1+70 are, therefore, special cases of Equations 1+65 and 1+66, respectively, where the yield function i is restricted hy Equation I+67. In the next section we will discuss the procedure hy which these equations can he utilized for specific yield functions. Prandtl-Reuss material 133. Prandtl-Reuss material is the most widely used, and perhaps the simplest, ideal elastic-plastic material. The yield condition asso ciated with the Prandtl-Reuss material is the well-known Von Mises cri terion given hy (V il) Equation 1+71 describes a right-circular cylinder in the principal stress space with its central axis the line of hydrostatic stress as shown in Figure l6. When the state of stress is such that Equation 1+71 is satis fied, the material would flow plastically, undergoing plastic as well as elastic strains. When the stresses are less than those satisfying Equa tion 1+71, the material will undergo elastic strains only. 13l+. In order to obtain the constitutive equation of Prandtl- Reuss material we simply substitute Equation 1+719 for the yield function 5 into Equation I+69. Completing the substitution, and considering the fact that during plastic deformation obtain S dE mn mn (bj2) 152 °2 Figure 16 . Von Mises yield surface in principal stress space Similarly, substituting Equation 1+71 in Equation 1+70 we obtain the fol lowing expression for stress increment tensor GS dE mn mn da.. = 2G dE.. + K dl_ 6.. S. . (1*73) ij ij 1 iJ ij The quantity dE in Equations 1+72 and 1+73 is recognized as the rate of work due to distortion. Expanding this quantity with respect to the plastic and elastic components we obtain S dE = S ( dEe + dEP (VrM mn mn mn \ mn mn) Since dE6 = dS /2G (see Equation l+13b), Equation I+7I* becomes mn mn S dS S dE + S dEP (1*75) mn mn 2G mn mn The quantity S dS is the increment of the second invariant of mn mn 153 stress deviation tensor (see Equation ¡+19) and is zero for the Von Mises yield criterion. Equation 1+75 reduces to S dE = S dEP mn mn mn mn (1+76) indicating that in the plastic range the rate of distortional work is only due to plastic deformation. Also, from Equations 1+72 and 1+73 it follows that d c . . d e , li li (1+77) In view of Equation 1+52, Equation 1+77 indicates that de?. = 0 (1+78) il That is, no plastic volume change can occur in the plastic range for Prandtl-Reuss material. 135. We can now summarize the Prandtl-Reuss equation in the fol lowing manner. During elastic loading m < and during unloading ( (9^/^aij)dff^j < o)» the elastic constitutive equation (Equation 1+53) prevails. In the plastic range m = k and (SjJ/Sa^ )da..j = 0^ , Equa tion 1+72 (or Equation 1+73) governs the behavior of the material. The Prandtl-Reuss constitutive equation can then be expressed as da. . = 2G dE. . + K dl, <5. . ij ij 1 ij when VjÂ" < k or d a . . < 0 (l+79a) 2 3a ij ij GS dE da. . = 2G dE. . + K dl, Ô. . - — ^ — — s ij 1 ij k2 ij when VJA = k and da. . = 0 2 3a.. ij (A79h) 136. In order to demonstrate the application of Equation 1+79, we will examine the behavior of Prandtl-Reuss material under conditions of 151+ uniaxial strain. For uniaxial strain conditions the strain increment and strain deviation increment tensors are given as de 0 0 de. . = 0 0 0 (U80a) ij 0 0 0 f - de 0 0 3 1 dE. . = 0 - k de. 0 (k80b) ij 3 1 1 0 0 den 3 In the elastic range the behavior of the material is governed by Equa tion ^79a. da1 = (k + J g ) d£l = M de1 = ^ (l+8la) on da, - da„ = da, - da0 = 2G de, = dJ, (U8lb) -L d 1 o I oJi -L For virgin loading in the elastic range, Equation h8l governs the be havior of the material. It should be noted that if the initial state of stress and strain is zero, for virgin loading, Equation l*8l can be used in terms of total rather than incremental quantities. In the uniaxial strain test (U82) Thus the material will yield when (cr - a ) = k (^8 3 ) V 3 155 In view of Equations bQl and 483, the value of vertical stress at yield becomes /3 (3K + 4g ) _ /3 M a (484) 1 6g k 2G k Thus, when reaches the value given by Equation 484, the material yields and continued application of vertical stress causes the material to move along the yield surface, undergoing plastic as well as elastic strains. In the plastic range Equation 479a no longer applies and re course to Equation 479b is necessary. According to Equation 479b, the deviator stress increment dS^ in the plastic range is given by GS dE dS1 = 2G dE1 - ■■■m2 11111 S (485) k The rate of work S dE for conditions of uniaxial strain reduces to mn mn S dE = S., dE., + 2S_ dE0 (486) mn mn 1 1 2 2 Utilizing the fact that = d E ^ = 0 Equation 486 reduces to S dE = -| S., dEn mn mn 2 1 1 (487) In view of Equations 1*8 Ob and U87* Equation U85 becomes _ 4g fi . (488) dSl 3 del ,2 del 2 — 3 2 In the plastic range k = ^ = "4 S1 and E dS = 0 (489) Since dS.„ = 0 , Equation 489 indicates that dS^ = 0 also, and d o± = dS^^ + dJx/3 = d^/3 (490a) da2 = dS2 + dJ1/3 = d^ / 3 (490b) 156 Equation 1+90 indicates that the material behaves as though it were a fluid once it has reached its limiting shear resistance. From Equa tions 1+T9h and l+8da, it follows that dJ^/3 = K de1 (*+9l) Substituting Equation 1+91 into Equation l+90a, the vertical stress-strain increment relation in the plastic range becomes da1 = K de1 (1+92) Thus, the loading slope of the versus curve breaks, or softens, when yielding occurs and becomes equal to the bulk modulus. Accordingly, the loading slopes of the principal stress difference-pressure curve and the principal stress difference-strain difference curve become zero. Since de^ = 0 , the slope of the pressure-volumetric strain curve kk remains constant. Once the material unloads, it behaves as a linear elastic solid again, satisfying Equation b8l. If unloading is continued until the lower yield surface corresponding to 1^ (o. - a 2 ) = - k (1+93) ¿3 is reached, the material flows plastically again and Equation l+79b governs the behavior of the material. The foregoing analyses are depicted schematically in Figure 17. From Figure 17 it can readily be seen that for a Prandtl-Reuss material, the vertical stress-strain curve associated with uniaxial strain configuration would break or soften when yielding occurs and would remain concave to the strain axis with continued application of vertical stress. 137. Let us next examine the behavior of Prandtl-Reuss material under a plane stress condition defined by a 0 0 1 0 0 0 (W 0 0 a 157 °i (0-j - a 2 ) a. VERTICAL STRESS-STRAIN RELATION b. PRINCIPAL STRESS DIFFERENCE- STRAIN DIFFERENCE RELATION (°1 —° 2 > c. PRESSURE-VOLUMETRIC STRAIN RELATION d. PRINCIPAL STRESS DIFFERENCE- PRESSURE RELATION ( STRESS PATH) Figure 17. Behavior of Prandtl-Reuss material under conditions of uniaxial strain 158 For this state of stress J 2 Vs) (495) Thus, the material "will yield when G1 + °3 - °la3 = 3k£ (496) Equation 496 describes an ellipse in the 0^ , 0^ coordinate system (Figure 18). We will consider a stress path where 0^ is held constant at k while 0^ is increased. At the start of the test (assume a compression test), point A in Figure 18, 0^ = 0 According to Equa tion 496 the material yields when 0^ = 2k , point B in Figure 18. Prior to yield the behavior of the material is governed by Equation 479a P ° i °l.el 159 9KG da E de^ 1 3K + G den (^97a) / 3K - 2G de den -V de^ (U9Tb) 3 y 6 K + 2G de2 = de3 (U97c) At point B the material yields and it follows from Equation ^59 that de^ = 0 (U98a) de| = -de^ (^9&b) Unlimited plastic deformation takes place at yield. It is noted from Equation H98 that, as expected, d e^ = 0 • If we now repeat the same test and change the direction of (i.e., a tension test), we find that the material yields when = -k , point C in Figure l8. At point C the material yields in tension and from Equation U59 it follows that II CO 0 (¡+99a) de^ = -de^ (^99b) The concept of normality can he demonstrated from this simple example hy superimposing the plastic strain coordinates on the stress coordinates in Figure 18. As shown in Figure 18, in the case of the compression test de^ = 0 and the plastic strain increment vector de^ is per pendicular to the yield surface at point B. In the case of the tension test, on the other hand, de^ = -de:^ indicating that the plastic strain increment vector is perpendicular to the yield surface at point C. Drucker-Prager material 138. The Von Mises yield condition was modified hy Drucker and 12 Prager to include the effects of the hydrostatic stress on the shear ing resistance of the material. The yield function $ was assumed to take the following form (500) 160 where a_^ ,9 a positive material constant, represents the frictional strength of the material. Equation 500 describes a right circular cone in the principal stress space (Figure 19)- Substituting Equation 500 ° 2 principal stress space into Equation U6 9 we obtain the following stress-strain relationship associated with the Drucker-Prager yield function — S dE - 3Ka dl, T mn mn f 1 d S . . dJn d e . . u ■ 1 2G + 9K 6ij + ij 9Kaf + G (501) t e " “ f 6 i j From Equation 501 it follows that l6l G , .__ S dE 3Ka„ dl, ' mn mn f 1 aett ■ -3v (502) 9Kaf + G indicating that for Drucker-Prager material, as a consequence of depen dency of yield function on hydrostatic stress, plastic deformation is accompanied by volume expansion (it is noted from Equation 502 that if = 0 the plastic volumetric strain is zero also). The increment of total volumetric strain dl1 can be determined from Equations 501 and 500. From Equation 501 we have (o de - J, dl., mn 1 1 n ) - 3Kaf dl dJ y j T V mn - 3a, (503) dIl = 3 F 9Kaf + G Solving for dl and considering the fact that during plastic deforma tion - = ^ (Equation 500), ¥e obtain V j | d J , . 3af ■(pKa^ + Gy - n a de dh - 3KGk k mn mn (50U) The increment of plastic volumetric strain de^ then becomes kk 3af dJ± „P ^ aJl/Q F 2 + r, -isKaij; + g) - askk = 3KGk (9Kaf + k amn demn 3K (505) It should he pointed out that the volume change is due to scalar non linearity and represents uniform dilatation. For example, consider a simple shearing stress defined by the folio-wing stress increment tensor Ö C\J 0 (— 1 0 da. . = 0 0 (506) ij d 0 0 0 From Equations 501 and 50U it follows that for this state of stress 162 del = d£2 = de3 (507) That is, there are no normal deviatoric strains associated with the volume expansion. 139* Substituting Equation 500 in Equation ^70 we obtain the following relationship for the stress increment tensor for the Drucker- Prager material = S dE - 3Ka„ dl, t T mn mn f 1 da. . = 2G dE.. + K dl_ 6. . - ij ij 1 iJ 9Kaf + G X 3KV i j (508) Equation 508 (or Equation 501) governs the behavior of Drucker-Prager material. The effect of the dependency of the yield function on hydro static stress can be further demonstrated by examining the behavior of Drucker-Prager material under uniaxial state of strain (Equation ii8o). The elastic behavior of the material is given by Equation l+8l. The mate rial yields when — (a - a_) - af (a1 + 2Cg) = k (509) /3 1 2 In view of Equations i*8l and 509» the value of vertical stress at yield becomes . _ ¿3 (3K + UG) L _ /3 Mk (51Q) 1 6G - 9/3 Kaf 2G - 3/3 Kaf It is noted that if af is set to zero, Equation 510 reduces to Equa tion U8U, the corresponding expression for Prandtl-Reuss material. The effect of otf in this case is to increase the value of the vertical stress a a t yield. When c1 reaches the value given by Equation 510, the material yields. Continued application of vertical stress causes the material to move along the yield surface, undergoing both elastic and plastic deformation. From Equation 508 the incremental relation be tween vertical stress and vertical strain becomes 163 da de 1 2 1 (511) 9Ka + G f Again it is noted that when is set to zero Equation 511 reduces to Equation ^92, the corresponding expression for Prandtl-Reuss material. As was pointed out previously, for Drucker-Prager material plastic de formation is accompanied "by volume expansion (see Equation 502). Accord ingly, using Equations 50U and 509, we obtain the following incremental relation for volumetric strain in the case of uniaxial strain test 9Ka 2 / 3 G - 3Ka, (■ dJl = dl1 + 3K dl1 (5 1 2 ) 9Ko^ + G When is set to zero, Equation 512 reduces to the corresponding expression for elastic material. The increment of plastic volumetric strain then becomes a (9Ka - 2 / 3 G) der_ = — ------dJ. (513) k 3KG(1 + 2/3 a ) In order for the uniaxial strain-stress path to reach the yield surface, the following condition should hold ~ ~ T ~ ~ > 3 a ( 5 1 * 0 r3 K Therefore, as expected, the increment of plastic volumetric strain is negative (expansion). Work-Hardening Plastic Material lUO. In the case of work-hardening plastic material, the yield surface l is not fixed but expands, or translates, as plastic defor mation takes place. The material can then sustain stresses beyond those required to reach the initial yield condition. Therefore, we can use a loading concept in the case of work-hardening plastic material according to the direction of the stress increment tensor da.. (viewed as vector). 1 6k During loading from a point on a given yield surface the stress vector is pointing outward and thus ( ) da^j > 0 During unloading the stress vector is pointing inward and thus ^ < 0 Accord- ij ingly for work-hardening plastic material we define da. . >0 loading (515a) 9°ij 1J da. . <0 unloading (515h) 9°ij 10 The condition (3^/3a..)da.. = 0 (i.e., when da., is tangent to yield ij ij surface) is known as neutral loading and produces no plastic deformation in the case of work-hardening material. The stability condition for work-hardening plastic material is given as da.. de.. > 0 (5l6a) ij ij da. . de? . > 0 (516b) ij iJ where, unlike the ideal plastic material, the equality sign in Equa tion 5l6b holds only when de?. = 0 . For work-hardening plastic mate- 11 1J rial Drucker has shown that the expression for plastic strain incre ment tensor is similar to Equation 1+59 "where the proportionality factor A depends on stress, plastic deformation, and history of plastic de formation. We can, therefore, use Equation 1+60, in conjunction with the loading conditions given by Equation 515, for calculating the strain increment tensor. During loading from a point on the yield surface (Oi/3a , .) da.. > 0), Equation ^60 governs the behavior of the material. f J J In the elastic range, and during unloading from a point on the yield surface ((3^/3a..) da.. < 0), the behavior of the material is governed by Equation ^53. When (8$/9a..)da.. = 0 (neutral loading), de?. = 0 and ij ij ij Equations 1+60 and 1+53 become identical (thus establishing continuity at a load-unload interface). ll+l. We now adopt a yield condition of the following type i = l(o. . , eP ) = ( ) u u y ij ’ mn/ 517 165 for strain-hardening material. Equation 517 indicates that the yield surface is not fixed in the principal space and that it changes as plastic deformation takes place. We further assume that k is a con stant. Following the same procedure as was used to derive an expression for A in the case of ideal plastic material we obtain . 3K - 2G drf de. . + — 7 7 --- dl, t — u_ 3a. . ij 6G 1 3a. ij A = __ (518) J j L _ 9 £ _ * 3K - 2G 3a. . 3a. . 6g 3a. . ij ij 3s?. ij ij Equation 518 is the expression for the proportionality factor A asso ciated with the strain-hardening yield condition given by Equation 517. It is noted that Equation 518 reduces to Equation 1+6U when the dependency of the yield function on the plastic strain disappears (i.e., ideal plastic material). In view of Equations 518 and U60, the strain incre ment tensor associated with the yield condition of Equation 517 becomes dS. . dJ a£ij= - # + 5 T 3j( 1 3K - 2G 3f( de ~ 0 3a mn + 6G dIl 3a mn mn mn H 3a. . (519) M. H + 3K - 2 G / H , f 3i( d{\ ij 3a 3 a 6G 3 a mn mn \ 9amn “ 7 ’ 3eP mn mn We can also derive an expression for the stress increment tensor do. . = 2G dE. . + K dl, _9jL de +... ___ 3K>~ 2G dl 6 3 a at'mn 6G 1 3a mn mn mn H H , 3K - 2G / 3rf ' 3a 3a 6G \ 3a mni mn mn \ mn / 3e^ ^amn mn 3K - 2G , 6 ) 6. . + 2G -r-^- 3 3a mn/ ij 3a. . (520) (; mn / ij 166 For a number of engineering materials, soils in particular, the yield function l is expressed in terms of , V j T > 2 ’ and ekk » 1*e' (521) For the above specification of ^ , Equations 519 and 520 become 3K dl + _G____ dE 1 3J. ~ T mn mn dS.. dJ. V j T 2 d e . -_3J . ^ —L 6 . . + ij 2G 9K ij 3i? 9K & ) + w - 3 kk aJi X /|f- 6.. + ~ ^ = z ^ = S.) (522) 13j i i j 2y^r ay3T i j . and 3K dl M_ + _ q _ S dE — . mn mn 1 9Ji V jI 3'VjI d a . . = 2G d E . . + K dl.. 6. . - i«J ij 1 ij 9K/li_\ + J j A — \ _ 3 _iL. M _ _9jL S.. (523) 9J i l j V j I s V j I 19 2 * 2 It is noted that Equations 522 and 523 reduce to Equations 1*69 and ^70, respectively, when the dependency of the yield function on disappears. ll+2. In order to demonstrate the application of Equation 522 (or Equation 523) let us consider an elliptic yield function defined by the following equation (Figure 20) 2 = o (52k) <(ji - vir, *) = - *) +Q Equation 52U has been used successfully for modeling the stress-strain 13 behavior of earth materials. For a first-order approximation, the variable Y , which controls the expansion of the yield surface, is assumed to take the form Y = A e ^ (525) 167 v% Figure 20. Work-hardening elliptic yield surface where A is a material constant which must he determined experimentally. In order to obtain the constitutive equation for the assumed work hardening yield surface we substitute Equation 52b9 for the yield sur face l , into Equation 522. Completing the substitution and consider ing the fact that d{) _ _3Y (526) P kk 9Y < kk w e o b t a i n 3KQ^(2J - y ) dl., + 2GS dE ____ V 1____ / 1______mn mn d e . . 6 . . + ij ij + + 3CfAJ (2J - Y SJ - i f UcJ'2 ) X (527) 168 lk3* Let us now examine the behavior of Equation 527 under hydro static state of stress (Figure 6a). For hydrostatic state of stress = Y and Equation 527 becomes (3K + A) dl dJ„ (528) 1 3KÂ It should be noted that for this state of stress the same results could have been obtained directly from Equations i+52, 1+53 9 and 525 without recourse to Equation 527* For virgin loading, Equation 528 can be in tegrated to yield (assuming zero initial pressure and volumetric strain) J K I_ (529) 1 e 1 During purely elastic deformation (Equation 1+53) dJ± = 3K (530) Since < 3K (Equation 528), it follows that plastic compaction pro- duces an apparent softening of the effective bulk modulus. Figure 21 depicts the behavior of the material under hydrostatic state of stress. The behavior of the material from point 1 to point 2 is governed by Equation 529 (the material undergoes plastic as well as elastic deforma tion). If the material is unloaded from point 2 to point 3 5 and then reloaded from point 3 to point 2, the behavior is elastic and the re sponse of the material is governed by Equation 530. ikk. Let us next examine the behavior of Equation 527 under a constant-pressure shear test (Figure 6c). The qualitative behavior of the model is depicted in Figure 22. The material is first hydro statically loaded from point 1 to point 2. The response of the material from point 1 to point 2 is governed by Equation 529 and is identical to that shown in Figure 21 (the material undergoes both plastic and elastic deformation). The material is then sheared from point 2 to point 3 by increasing n while is kept constant. Since is kept con stant, all volume changes from point 2 to point 3 are plastic. From 169 Figure 21. Behavior of -work-hardening elastic-plastic material under hydro static state of stress 170 Vi Vi Figure 22, Behavior of work-hardening elastic-plastic material during constant-pressure shear test Equation ^59 it follows that the increment of plastic volumetric strain is given as d i = 3A|f- (531) kk dJ^ In view of Equation 52^, Equation 531 becomes dekk = 3^(2Ji - Y) (532) Since A is positive, Equation 532 indicates that the plastic volumet ric strain during the shearing process is positive (compaction). At point 3 = 0 (normality condition) and the yield surface ceases to expand. The shearing response of material, expressed in terms of versus V i J , then reaches its maximum value (for the particular value 171 of at point 2) asymptotically at point 3. As shown by the dashed lines in Figure 22, if the material were to unload from any point during the shearing process it will behave as a linear elastic material. This simple example points out the basic difference between ideal and work hardening plastic materials. That is, for ideal plastic materials the yield surface is fixed and does not expand during plastic deformation. Unlimited plastic flow takes place at the onset of yielding. In the case of work-hardening material, on the other hand, the yield surface moves, or expands, causing the material to harden as plastic deformation takes place. 172 REFERENCES 1 . Rivlin, R. S. and Ericksen, J. L., "Stress-Deformation Relation for Isotropic Materials," Journal, Rational Mechanics and Analysis, Vol U, No. 2, Mar 1955, PP 323-^25- 2. Harr, M. E., Foundations of Theoretical Soil Mechanics, 1st ed., McGraw-Hill, New York, 1966. 3. Kami, Z. and Reiner, M., "The General Measure of Deformation," Second-Order Effects in Elasticity, Plasticity and Fluid Dynamics, M. Reiner and D. Abir, ed., Macmillan, New York, 196b, pp 217-227. 4. Eringen, A. C., Nonlinear Theory of Continuous Media, McGraw-Hill, New York, 1962. 5. Langharr, H. L., Energy Methods in Applied Mechanics, Wiley, New York, 1962. 6 . Evans, R. J. and Pister, K. S., "Constitutive Equations for a Class of Nonlinear Elastic Solids," International Journal of Solids and Structures, Vol 2, No. 3, Jul 1966, pp h2J-bh5- 7. Nelson, I. and Baron, M. L., "Investigation of Ground Shock Effects in Nonlinear Hysteretic Media; Development of Mathematical Material Models," Contract Report S-68-1, Report 1, Mar 1968, U. S. Army Engineer Waterways Experiment Station, CE, Vicksburg, Miss.; pre pared by Paul Weidlinger, Consulting Engineer, New York, under Contract No. DACA39-67-C-00i|8. 8 . Nelson, I., "Investigation of Ground Shock Effects in Nonlinear Hysteretic Media; Modeling the Behavior of a Real Soil," Contract Report S-68-1, Report 2, Jul 1970, U. S. Army Engineer Waterways Experiment Station, CE, Vicksburg, Miss.; prepared by Paul Weid linger, Consulting Engineer, New York, under Contract No. DACA39- 67-C-OOU8 . 9. Prager, W., "Theory of Plastic Flow Versus Theory of Plastic Deformation," Journal, Applied Physics, Vol 19» No. 6, Jun 19^.8, pp 5^0-5^3. 10. Flugge, W., Viscoelasticity, Blaisdell, Waltham, 1967. 11. Drucker, D. C., "A More Fundamental Approach to Plastic Stress- Strain Relations," Proceedings, First U. S. National Congress on Applied Mechanics, 1951» PP ^87-^91* 12. Drucker, D. C. and Prager, W., "Soil Mechanics and Plastic Analysis or Limit Design," Quarterly of Applied Mathematics, Vol 10, No. 2, Jul 1952, pp 157-175. 13. Roscoe, K. H. and Burland, J. B., "On the Generalized Stress-Strain Behavior of ’Wet' Clay," Engineering Plasticity, J. Heyman and F. A. Leckie, ed., Cambridge University Press, 1968, pp 535-609- 173 APPENDIX A: SELECTED BIBLIOGRAPHY This appendix contains 78 references that are related to subject matter presented in the main text of this report. These references are in addition to those listed following the main text and are given in alphabetical order. For easy reference and informational purposes, the references in this appendix are cataloged in Table A1 in accordance with their relation to three basic material models: incremental models, plasticity, and viscoelasticity. A1 BIBLIOGRAPHY Adachi, T ., Construction of Continuum Theories for Rock by Tensor Testing, Ph. D. Dissertation, University of California, Berkeley, Calif., 1970. Adler, F. T., Sawyer, W. M., and Ferry, J. D., ’’Propagation of Trans verse Waves in Viscoelastic Media,” Journal, Applied Physics, Vol 20, No. 11, 19^9, pp 1036-101+1. Alfrey, T., ”Non-Homogeneous Stresses in Viscoelestic Media,” Quarterly of Applied Mathematics, Vol 2, 19I+J+, pp 113-119« Alfrey, T. and Doty, P., ’’Methods of Specifying the Properties of Visco elastic Materials,” Journal, Applied Physics, Vol l6, No. 11, 19^+5, pp 700-713, Baker, L. S. and Kesler, C. E., ”A Study of the Damping Properties of Mortar,” Report 598, Aug 1961, Department of Theoretical and Applied Mechanics, University of Illinois, Urbana, 111. Baron, M. L., McCormick, J. M. , and Nelson, I., ’’Investigation of Ground- Shock Effects in Non-Linear Hysteretic Media,” Computational Approaches in Applied Mechanics, American Society of Mechanical Engineers, Jun 1969, pp 165-190. Biot, M. A., ’’Theory of Stress-Strain Relations in Anisotropic Visco elasticity and Relaxation Phenomena,” Journal, Applied Physics, Vol 25, No. 11, 195*+, PP 1385-1391. ______, ’’Dynamics of Viscoelastic Anisotropic Media,” Proceedings, Second Midwestern Conference on Solid Mechanics, Purdue University Engi neering Experiment Station Research Series No. 129, 1956a, pp 9^+-108. ______, ’’Theory of Deformation of a Porous Viscoelastic Anisotropic Solid,” Journal, Applied Physics, Vol 27, No. 5, 1956b, pp J+59-^+67« Bland, D. R., The Theory of Linear Viscoelasticity, Pergamon, New York, I960. Bland, D. R. and Lee, E. H., "Calculation of the Complex Modulus of Linear Viscoelastic Materials from Vibrating Reed Measurements," Journal, Applied Physics, Vol 26, No. 12, 1955, PP 11+97-1503« Bleich, H. H., "On the Use of a Special Nonassociated Flow Rule for Problems of Elasto-Plastic Wave Propagation,” DASA 2635, Mar 1971, Defense Atomic Support Agency (Defense Nuclear Agency), New York, N. Y. Bleich, H. H. and Heer, E., "Moving Step Load on a Half-Space of Granular Material,” Journal, Engineering Mechanics Division, American Society of Civil Engineers, Vol 89, No. EM3, Jun 1963, pp 97-129. Bleich, H. H. and Matthews, A. T., "Exponentially Decaying Pressure Pulse Moving with Superseismic Velocity on the Surface of a Half-Space of Granular Material,” Technical Report No. AFWL-TR-67-21, Jul 1967a, Air Force Weapons Laboratory, Albuquerque, N. Mex. A2 Bleich, H. H. and Matthews, A. T., "Step Load Moving with Superseismic Velocity on the Surface of an Elastic-Plastic Half-Space," International Journal of Solids and Structures, Vol 3, No. 5, Sep 196Tb, pp 819-852. ______, "Exponentially Decaying Pressure Pulse Moving with Super- seismic Velocity on the Surface of a Half-Space of a von Mises Elasto- Plastic Material," Technical Report No. AFWL-TR-68-1+6, Aug 1968, Air Force Weapons Laboratory, Albuquerque, N. Mex. Bleich, H. H., Matthews, A. T., and Wright, J. P., "Step Load Moving with Superseismic Velocity on the Surface of a Half-Space of Granular Material," Technical Report No. AFWL-TR-65-59, Sep 1965, Air Force Weapons Laboratory, Albuquerque, N. Mex. ______, "Moving Step Load on the Surface of a Half-Space of Granular Material," International Journal of Solids and Structures, Vol 1+, No. 2, Feb 1968, pp 21+3-286. Bodner, S. R . , "Strain Rate Effects in Dynamic Loading of Structures," Behavior of Materials Under Dynamic Loading, American Society of Mechanical Engineers, N. J. Huffington, ed., 1965, PP 93-105. Bridgman, P. W . , "Volume Changes in the Plastic Stages of Simple Com pression," Journal, Applied Physics, Vol 20, No. 12, Dec 19^9, PP 121+1- 1251. Brown, W. S. and Swanson, S. R., "Constitutive Equations for Rocks Based on Experimental Measurements for Stress-Strain and Fracture Properties," presented at the DASA Long Range Planning Meeting, Albuquerque, N. Mex., 1970. Christian, J. T., "Two-Dimensional Analysis of Stress and Strain in Soils; Plane-Strain Deformation Analysis of Soil," Contract Report No. 3-129, Report 3, Dec 1966, U. S. Army Engineer Waterways Experiment Station, CE, Vicksburg, Miss.; prepared by Massachusetts Institute of Technology, under Contract No. DA-22-079-eng-l+71. Coleman, B. D. , "Thermodynamics of Materials with Memory," Archive for Rational Mechanics and Analysis, Vol 17, No. 1, 1961+, pp 1-1+6. Cristescu, N., Dyanmic Plasticity, American Elsevier, New York, 1967. DeWitt, T. W . , "A Rheological Equation of State Which Predicts Non- Newtonian Viscosity, Normal Stresses, and Dynamic Moduli," Journal, Applied Physics, Vol 26, No. 7, 1955, pp 889-89I+. Drucker, D. C., "On Uniqueness in the Theory of Plasticity," Quarterly of Applied Mathematics, Vol ll+, 1956, pp 35-1+2. ______, "Variational Principles in the Mathematical Theory of Plasticity.," Proceedings, Symposium in Applied Mathematics, Vol 8, 1958, pp 7-22. , "A Definition of Stable Inelastic Material," Journal, Applied Mechanics, Vol 26, No. 1, Mar 1959, PP 101-106. A3 Drucker, D. C., "Survey on Second-Order Plasticity," Second-Order Effects in Elasticity, Plasticity, and Fluid Dynamics, M. Reiner and D. Abir, eds., Macmillan, New York, 196k, pp 1+16-1+23. ______, "The Continuum Theory of Plasticity on the Macroscale and the Microscale," Journal of Materials, Vol 1 , No. 1+, Dec 1966, pp 873-910. Drucker, D. C. and White, G. N., Jr., "Effective Stress and Effective Strain in Relation to Stress Theories of Plasticity," Journal, Applied Physics, Vol 21, 1950, pp 1013-1021. Drucker, D. C., Gibson, R. E., and Henkel, D. J., "Soil Mechanics and Work-Hardening Theories of Plasticity," Transactions, American Society of Civil Engineers, Vol 122, 1957, pp 338-31+6. Finnie, I. and Heller, W. R., Creep of Engineering Materials, McGraw- Hill, New York, 1959- Freudenthal, A. M., The Inelastic Behavior of Engineering Materials and Structures, Wiley, New York, 1950. Freudenthal, A. M. and Geiringer, H., "The Mathematical Theories of the Inelastic Continuum," Handbuch der Physik, VI, S. Flugge, ed., Springer- Verlag, Berlin, 1958. Fung, Y. C., Foundation of Solid Mechanics, Prentice—Hall, Englewood Cliffs, 1965. Green, A. E. and Naghdi, P. M., "A General Theory of an Elastic-Plastic Continuum," Archive for Rational Mechanics and Analysis, Vol 18, 1965, pp 251-281. Gross, B., Mathematical Structure of the Theories of Viscoelasticity, Hermann, Paris, 1953. Halpin, J. C., "Introduction to Viscoelasticity," Composite Materials Workshop, S. W. Tsai, J. C. Halpin, and N. J. Pagano, eds., 1968, pp 87-152. Hill, R., The Mathematical Theory of Plasticity, Oxford University Press, London, 1950. ______, "A General Theory of Uniqueness and Stability in Elastic- Plastic Solids," Journal, Mechanics and Physics of Solids, Vol 6 , No. 3, May 1958, pp 236-21+9. Huang, N. C. and Lee, E. H., "Nonlinear Viscoelasticity for Short Time Ranges," Journal, Applied Mechanics, Vol 33, Jun 1966, pp 313-321. Il'yushin, A. A., "On the Postulate of Plasticity," Prikladnaya Matematika i Mekhanika, Vol 25, No. 3, May-Jun 1961, pp 503-507 ("0 Postulate Plastichnosti," Translation, Journal, Applied Mathematics and Mechanics, Vol 25, No. 3, 1961, pp 71+6-762). Jenike, A. W. and Shield, R. T., "On the Plastic Flow of Coulomb Solids Beyond Original Failure," Journal, Applied Mechanics, Vol 26, Dec 1959, PP 599-602. Konder, R. L. and Krizek, R. J., "A Rheologic Investigation of the Dynamic Response Spectra of Soils; Basic Concepts, Equipment Develop ment, and Soil Testing Procedures,” Contract Report No. 3-62, Report 1, Dec 1962, U. S. Army Engineer Waterways Experiment Station, CE, Vicks burg, Miss.; prepared by Northwestern University, under Contract No. DA-22-079-eng-3l6. ______, "A Rheologic Investigation of the Dynamic Response Spectra of Soils; A Response Spectra Formulation for a Cohesive Soil," Contract Report No. 3-62, Report 2, Aug 1963, U. S. Army Engineer Waterways Ex periment Station, CE, Vicksburg, Miss.; prepared by Northwestern Uni versity, under Contract No. DA-22-079-eng-3l6. Lee, E. H., "Stress Analysis in Viscoelastic Bodies," Quarterly of Applied Mathematics, Vol 13, Jul 1955, PP 183-190. Lee, E. H. and Radok, J. R. M . , "Stress Analysis in Linearly Visco elastic Materials," Proceedings, Ninth International Congress for Applied Mechanics, Vol 5, 1957, PP 321-329. Malvern, L. E., Introduction to the Mechanics of a Continuous Medium, Prentice-Hall, Englewood Cliffs, 1969. Matthews, A. T. and Bleich, H. H., "Investigation of Ground Shock Effects in Nonlinear Hysteretic Media; A Note on the Plane Waves of Pressure and Shear in a Half-Space Hysteretic Material," Contract Report S-68-1, Report 3, Jan 1969, U. S. Army Engineer Waterways Experiment Station, CE, Vicksburg, Miss.; prepared by Paul Weidlinger, Consulting Engineer, New York, under Contract No. DACA39-67-C-001+8. Perzyna, P., "The Constitutive Equations for Rate Sensitive Plastic Materials," Quarterly of Applied Mathematics, Vol 20, No. U, Jan 1963, pp 321-332. Prager, W . , "Strain Hardening Under Combined Stresses," Journal, Applied Physics, Vol 16, No. 12, 19 U5 , PP 837-81+0. ______, An Introduction to Plasticity, Addison-Wesley, Reading, 1959. Radok, J. R. M . , "Viscoelastic Stress Analysis," Quarterly of Applied Mathematics, Vol 15, 1957, PP 198-202. ______, Introduction to Mechanics of Continua, Ginn, Boston, 19 6 1. Read, W. T., Jr., "Stress Analysis for Compressible Viscoelastic Mate rials," Journal, Applied Physics, Vol 21, 1950, pp 67I-67I+. Reiner, M . , Twelve Lectures on Theoretical Rheology, Interscience, New York, 19 I+9. , Deformation, Strain and Flow, 2d ed., Interscience, New York, 1960a. ______"Plastic Yielding in Anelasticity," Journal, Mechanics and Physics of Solids, Vol 8 , No. k 9 Nov 1960b, pp 255-261. Reiner, M. and Abir, D., ed. , Second-Order Effects in Elasticity,- Plasticity, and Fluid Dynamics, Macmillan, New York, 1961+. A5 Rivlin, R. S., "Nonlinear Viscoelastic Solids/1 Society for Industrial and Applied Mechanics, Vol 7, Jul 1965, pp 323-3^0. Schapery, R. A, "Irreversible Thermodynamics and Variational Principles with Applications to Viscoelasticity," ARL 62-Ul8, 1962, Aeronautical Research Laboratory, Wright-Patterson Air Force Base, Ohio. ______, "An Engineering Theory of Nonlinear Viscoelasticity with Applications," Jun 1965, p 31, School of Aeronautics, Astronautics, and Engineering Sciences, Purdue University, Lafayette, Ind. Schiffman, R. L., "The Use of Viscoe-Elastic Stress-Strain Laws in Soil Testing," Papers on Soils, 1959 Meetings, Special Technical Publica tion 25U, pp 131-1555 I960, American Society for Testing and Materials, Philadelphia, Pa. Schmid, ¥. E., Klausner, Y., and Whitmore, C. F., "Rheological Shear and Consolidation Behavior of Clay Soils," Technical Report No. NR-081-117, Dec 19699 Department of Civil Engineering, Princeton University, Princeton, N. J. Scott-Blair, G. W., A Survey of General and Applied Rheology, Pitman and Sons, London, 19^. Stroganov, A. S., "Plane Plastic Deformation of Soil," Conference on Earth Pressure Problems, Belgium Symposium of the International Society of Soil Mechanics and Foundation Engineering, Brussels, Vol 1, 1958, pp 9U-IO3 . ______, "Sur Certains Problemes Rheologiques de la Mecanique des Sols," Rheology and Soil Mechanics Symposium, Springer-Verlag, I96U, pp 237-250. Swanson, S. R., Development of Constitutive Equations for Rocks, Ph. D. Dissertation, Department of Mechanical Engineering, University of Utah, Salt Lake City, Utah, Jun 1970. Symonds, P. S., "Viscoplastic Behavior in Response of Structures to Dynamic Loading," Behavior of Materials Under Dynamic Loading, American Society of Mechanical Engineers, N. J. Huffington, Jr., ed., 1965, pp 106-129. Thomas, T. Y., "On the Structure of the Stress-Strain Relations," Proceedings, National Academy of Sciences, Vol ^1, No. 10, Oct 1955, pp 716-720. . ______, "Isotropic Materials Whose Deformation and Distortion Ener gies Are Expressible by Scalar Invariants," Proceedings, National Academy of Sciences, Vol ^2, No. 9, Sep 1956, pp 603-608. ______, "Deformation Energy and the Stress-Strain Relations for Isotropic Materials," Journal, Mathematics and Physics, Vol 35, No. Jan 1957, PP 335-350. ______, Plastic Flow and Fracture in Solids, Academic Press, New York, 1961. A 6 Truesdell, C., ”Hypo-Elastic Shear,” Journal, Applied Physics, Vol 27, No. 5, 1956, pp Ul-U*7. ______, ”Second-Order Effects in the Mechanics of Materials,” Second- Order Effects in Elasticity, Plasticity, and Fluid Dynamics, M. Reiner and D. Abir, eds., Macmillan, New York, 196^, PP 1-^7« Truesdell, C. and Noll, W . , ”The Nonlinear Field Theory of Mechanics,” Encyclopedia of Physics, S. Flugge, ed., Vol 3, No. 2, Springer-Verlag, 196^, pp U—5• Volterra, E., ”0n Elastic Continua with Hereditary Characteristics,” Journal, Applied Mechanics, Vol 18, 1951, PP 273-279« Weidler, J. B. and Paslay, P. R., Analytical Description of Behavior of Granular Media., Journal, Engineering Mechanics Division, American Society of Civil Engineers, Vol 95, EM2, Apr 1969, p p 379-395. A7 Table Al A Catalog of the Selected Bibliography Viscoelasticity Plasticity Incremental Models Adachi, 1970 Adachi, 1970 Thomas, 1955 Adler, Sawyer, and Baron, McCormick, and Thomas, 1956 Ferry, 19^+9 Nelson, 1969 Thomas, 1957 Alfrey, 1 9 ^ Bleich, 1971 Truesdell, 1956 Alfrey and Doty, 19^5 Bleich and Heer, 1963 Truesdell, I 96I+ Baker and Kesler, 1961 Bleich and Matthews, 1967a Truesdell and Biot, 195^ Bleich and Matthews, 1967h Noll, 1961+ Biot, 1956a Bleich and Matthews, 1968 Biot, 1956b Bleich, Matthews, and Bland, I960 Wright, 1965 Bland and Lee, 1955 Bleich, Matthews, and Bodner, 1965 Wright, 1968 Coleman, I 96U Bridgman, 19^-9 Cristescu, 1967 Brown and Swanson, 1970 Dewitt, 1955 Christian, 1966 Finnie and Heller, 1959 Cristescu, 1967 Freudenthal, 1950 Drucker, 1956 Fung, 1965 Drucker, 1958 Gross, 1953 Drucker, 1959 Halpin, 1968 D r u cker, 196I+ Huang and Lee, 1966 Drucker, 1966 Kondner and Krizek, 1962 Drucker and White, 1950 Kondner and Krizek, 1963 Drucker, Gibson, and Lee, 1955 Henkel, 1957 Lee and Radok, 1957 Freudenthal, 1950 Malvern, 1969 Freudenthal and Perzyna, 1963 Geiringer, 1958 Prager, 1961. Fung, 1965 Radok, 1957 Green and Naghdi, 1965 Read, 1950 Hill, 1950 Reiner, 19^-9 Hill, 1958 Reiner, 1960a Il’yushin, 1961 Reiner and Abir, 196I* Jenike and Shield, 1959 Rivlin, 1965 Malvern, 1969 Schapery, 1962 Matthews and Bleich, 1969 Schapery, 1965 Prager, 19U 5 Schiffman, i 960 Prager, 1959 Schmid, Klausner, and Prager, 1961 Whitmore, 1969 Reiner, 1960b Scott-Blair, 19I+I+ Stroganov, 1958 Stroganov, 196b Swanson, 1970 Symonds, 1965 T h o m a s , 1961 Volterra, 1951 Weidler and Paslay, 1969 A8 In accordance with letter from DAEN-RDC* DAEN-ASI dated 22 July 1977, Subject: Facsimile Catalog Cards for Laboratory Technical Publications, a facsimile catalog card in Library of Congress MARC format is reproduced below. Rohani, Behzad Mechanical constitutive models for engineering materials / by Behzad Rohani. Vicksburg, Miss. : U. S. Waterways Experiment Station ; Springfield, Va. : available from National Technical Information Service, 1977. 173, 8 p. : ill. ; 27 cm. (Miscellaneous paper - Ü. S. Army Engineer Waterways Experiment Station ; S-77-19) Prepared for Assistant Secretary of the Army (R&D), Depart ment of the Army, Washington, D. C., under Project 4A161101A91D. Appendix A: Selected bibliography. References: p. 173. 1. Constitutive equations. 2. Constitutive models. 3. Elastic media. 4. Plastic media. 5. Viscoelastic media. I. United States. Assistant Secretary of the Army (Research and Development). II. Series: United States. Waterways Experiment Station, Vicksburg, Miss. Miscellaneous paper ; S-77-19. TA7.W34m no.S-77-19