* " TECHNICAL REPORT SL-82-10 AN ELASTIC-VISCOPLASTIC CONSTITUTIVE MODEL FOR EARTH MATERIALS by C9, George Y. Baladi and Behzad Rohani Structures Laboratory U. S. Army Engineer Waterways Experiment Station P. 0. Box 631, Vicksburg, Miss. 39180 December 1982 Final Report Approvd For Public Rem Dltrb~~s Unlmited Prped for Office, Chief of Engineers, U. S. Army Washington, D. C. 20314 Under Project 4A161102AT22, Task BO, Work Unit 00623 1 4
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AN ELASTIC-VISCOPLASTIC CONSTITUTIVE MODEL FOR …mental elastic-plastic constitutive model for earth materials and intro--duce rate dependency in such a model. Generally, two different
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* "
TECHNICAL REPORT SL-82-10
AN ELASTIC-VISCOPLASTIC CONSTITUTIVEMODEL FOR EARTH MATERIALS
by
C9, George Y. Baladi and Behzad Rohani
Structures LaboratoryU. S. Army Engineer Waterways Experiment Station
P. 0. Box 631, Vicksburg, Miss. 39180
December 1982Final Report
Approvd For Public Rem Dltrb~~s Unlmited
Prped for Office, Chief of Engineers, U. S. ArmyWashington, D. C. 20314
Under Project 4A161102AT22, Task BO, Work Unit 006231 4
Destroy Ohin report when no longer aeemd Do eroe,.iIt to the originator.
The findings In this report are not to be construed as an officialDepartment of the Army position unless so designated.
by other authorited documents.
FM.USI ~Vde4
UnclassifiedMECUTY CLASSIICATION OP THIS PAGE (1110M DOS 00-94
Technical Report SL-82-10L~-4~ 3TITLE (4* 6"00~ S. TYPE OF REPORT 6 PERIOD COVERESD
AN ELABTIC-VISCOPLABTIC CONSTITUTIVE MODEL FOR FnlrprRART MATERIALS S. PERFORMING ORG. REPORT NUMBER
1. AUTVIM4 9. CONTRACT OR GRANT NUMUERW.)
George Y. laladi and Behzad Rohani
9 . PgnPOUND ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT. TASKAREA 6 WORK UNIT NUMBERS
U. S. Army Engineer Waterways Experiment Station Poet4110A2Structures LaboratoryP. 0. Box 631, Vicksburg, Miss. 39180 Ts O okUi 0
19. CONTROLLING OFFICE MNM AND ADDRESS 12. REPORT DATE .
December 1982* ~~Office, Chief of Engineers, U. S. Army SNMEO AEWashington, D. C. 20314 3
M4 ISOUITOIN AGENCY N-AME A AOORESU(H 41IN~et bA= Cant~I91111 O10160) 15. SECURITY CLASS. (et thi. sre
Unclassified
I"S..DCASSIFtiCATON/DOWNGRADING
) .51. DISTRISUTIOM STATEMENT (Of1 two hport)
Approved for public release; distribution unlimited.
17. DISTRIBUTION STATEMENT (6t t. abaoroW Mised in 8100k 20. it IffhernI &Mr R6ped)
IL. SUPPLEMENTARY NOTES
4 Available from National Technical Information Service, 5285 Port Royal Road,Springfield, Va. 22151.
IS. Key WORDM (Cedmou.an i~rom ofif it aeoear and IdetlY by bleak mnk..~)
Elastic-viscoplastic constitutive relationshipRate-dependent material propertiesSoil constitutive modelsStress-strain relations
SL ABSTRACT (Coeajbu a veywge 40d of neseear mod Ideni5fy by b10a uinb")
It has long been recognized that the stress-strain-strength properties of4most geological materials under high-intensity dynamic loading conditions differconsiderably from those related to relatively low-intensity static loadings.Thus realistic solutions to soil dynamic boundary-value problems require the useof constitutive models that can account for the dependency of the stress-strain-strength properties of earth materials to the various rates of loading ordeformation being applied. (Continued)
w I 47 F.ED"" cTioN oF 1 Nov s isa OSSOLETE UnclassifiedSECURITY CLASSFICATION OF THIS PAGE (Mma Dwa Rnted
1777 . 77
Unclassifled
*20. AUTNACT (Continued)
This report describes the development oA a three-dimensional elastic-viscoplastic, work-hardening constitutive relationship for earth materials.The constitutive relationship is capable of reproducing the hysteretic behaviorof the material under both hydrostatic and deviatoric states of stress; it alsoaccounts for shear-induced volume change and the effect of superimposedhydrostatic stress on shearing response. The capability of the constitutive -
relationship for simulating the time-dependent response of earth materials isexamned anexamle it or claey andis gvenbasd o staic bqrto1
triaxial shear and static and dynamic uniaxial strain test results
.. ~~4 . .. .. ..
PREFACE
This investigation was conducted for the Office, Chief of Engineers,
U. S. Army, by personnel of the Geomechanics Division (GD), Structures
Laboratory (SL), U. S. Army Engineer Waterways Experiment Station (WES),
as a part of Project 4A161102AT22, Task BO, Work Unit 005, "Constitutive
Properties for Natural Earth and Man-Made Materials." This report was
prepared by Drs. George Y. Baladi and Behzad Rohani during the period
January-July 1982 under the general direction of Mr. Bryant Mather,
Chief, SL, and Dr. J. G. Jackson, Jr., Chief, GD. The report was typed
by Mrs. Bobbie B. Morrow.
COL Tilford C. Creel, CE, was Commander and Director of WES during
.this investigation. Mr. F. R.Brown was Technical Director.
devPYS L A BO L/Rlde- _ 1 Yij jB J1 - )1 2 + -22('
dt 2 # J1 - L)2/R2 + J2
(49)
Equations 45 through 49 provide a complete specification for the behav- '
ior of the material for constant mean normal stress triaxial tests.
These equations, however, must be integrated numerically in order to
relate stresses to strains during dynamic loading. A computer program
has been developed for numerical integration of the governing equations rof the proposed model for general three-dimensional states of stress.
For the sake of brevity, the numerical implementation of the model is .
not described herein. In the following section, the behavior of the
model is correlated with test data for a clayey sand using this computer .
program.
26
S - - - ---.7
PART V: CORRELATION OF TEST DATA WITH MODEL BEHAVIOR
22. In this section, the behavior of the model under states of -
uniaxial strain and triaxial shear are correlated with available test
data for a clayey sand classified as SC according to the Unified Soil
Classification System. Static data consisted of (a) load-unload axial
stress-axial strain relations (a versus e ) for uniaxial strain, andz zthe corresponding stress paths expressed in terms of principal stressdifference versus mean normal stress (a - ar versus P) , and (b) two
z rtriaxial shear test load-unload stress-strain relations (for two
different confining stresses) presented in terms of principal stress |
difference versus principal strain difference (a - ar versusz r
Cz - £r) ,and a static failure envelope based on these two testsz r(Ehrgott, 1978). The available dynamic data for this material consisted
of several stress-strain curves from dynamic uniaxial strain tests
(Jackson, Ehrgott, and Rohani; 1980).
23. The first step in correlating the behavior of the model with
test data is to simulate the static properties of the material and to
determine the numerical values of the material constants Ki , K1 , I -
K2 , Gi, G1 , 2 , a , k ,R , R1 , R2 , D , and W The
response of the material under static loading is governed by these
constants (i.e., the case of inviscid plasticity when rate dependency is
neglected). Figure 11 portrays the static test data and the correspond-
ing model behavior for both the triaxial shear and the uniaxial strain
test conditions. It can be noted from Figure 11 that the model has
reasonably simulated the static response of the soil both qualitatively
(triaxial shear response) and quantitatively (uniaxial strain response).
The next step is to simulate the available dynamic stress-strain
properties of the material, using the numerical values of the 13 con-
stants above and numerical values determined for the remaining three
constants B , B1 , and y . As indicated before, the available
dynamic data for this material are limited to several uniaxial strain
stress-strain relations. The dynamic data were obtained for loading
25. An elastic-viscoplastic constitutive model has been developed
for earth materials and has been partially validated via comparison with
both static and limited dynamic stress-strain data for a clayey sand.
The model is capable of simulating many pertinent features of the
stress-strain-strength properties of earth materials such as dependency
of the shearing strength of the material on hydrostatic stress and rate
of deformation, shear-induced volume change, and permanent deformation
under hydrostatic and deviatoric cyclic loadings. In its present form,
* the model contains 16 material constants that can be readily determined
from the results of static and dynamic triaxial shear and uniaxial . .
strain tests.
26. The model has been translated into a numerical algorithm for
implementation into finite-difference or finite-element computer codes.
The numerical algorithm is very versatile in that it embodies all
classes of elastic-plastic constitutive models. Test data for several
rates of deformation and test boundary conditions (other than those used
to fit the model) are needed to further validate the accuracy and deter-
mine the range of application of the model.
33
- - -- -- -"- -_
REFERENCES
Raladi, G. Y. August 1977. "Numerical Implementation of a Transverse-Isotropic Inelastic, Work-Hardening Constitutive Model," Transactions ofthe 4th International Conference on Structural Mechanics in ReactorTechnology, Vol. M, Methods for Structural Analysis, San Francisco,Calif.
Baladi, G. Y. and Rohani, B. April 1979. "Elastic-Plastic Model forSaturated Sand," Journal of the Geotechnical Engineering Division, ASCE,Vol. 105, No. GT4, Proc. Paper 14510, pp. 465-480.
DiMaggio, F. L. and Sandler, I. October 1971. "The Effect of StrainRate on the Constitutive Equations of Rocks," Technical Report DNA2801T, Defense Nuclear Agency, Washington, D. C.
Drucker, D. C. 1956. "On Uniqueness in the Theory of Plasticity,"Quarterly of Applied Mathematics, Vol. 14.
Ehrgott, J. Q. 1978. "Loading Response of a Backfill Along FourDifferent Stress Paths," Internal Data Report, U. S. Army EngineerWaterways Experiment Station, CE, Vicksburg, Miss.
Jackson, J. G., Jr., Ehrgott, J. Q., and Rohani, B. August 1980."Loading Rate Effects on Compressibility of Sand," Journal of theGeotechnical Engineering Division, ASCE, Vol. 106, No. GT8, Proc. Paper15640, pp. 839-852.
Perzyna, P. 1966. "Fundamental Problems in Viscoplasticity," Advances
'" in Applied Mechanics, Vol. 9, Academic Press, New York, pp. 243-377.
Sandler, I. S., DiMaggio, F. L., and Baladi, G. Y. July 1976."Generalized Cap Model for Geological Materials," Journal of theGeotechnical Engineering Division, ASCE, Vol. 102, No. GT7, Proc. Paper12243, pp. 683-699.
Swift, R. S. July 1975. "Examination of the Mechanical Properties for
a Kayenta Sandstone from the MIXED COMPANY Site," Technical Report DNA3683F, Defense Nuclear Agency, Washington, D. C.
Whitman, R. V. May 1970. "The Response of Soils to Dynamic Loadings;Final Report," Contract Report No. 3-26, Report 26, U. S. Army EngineerWaterways Experiment Station, CE, Vicksburg, Miss.
34
k % %. , / • . .. -. . °.4
In accordance with letter from DAEN-RDC, DAEN-ASJ dated '
22 July 1977, Subject: Facsimile Catalog Cards for
Laboratory Technical Publications, A facsimile catalog ... -
card in Library of Congress MARC format is reproducedbelow.
Baladi, George Y.
An elastic-viscoplastic constitutive model for earth
materials / by George Y. Baladi and Behzad Rohani
(Structures Laboratory, U.S. Army Engineer Waterways
Experiment Station). -- Vicksburg, Miss. : The Station ; ."
Springfield, Va. : available from NTIS, 1982. -
34 p. : ill. ; 27 cm. -- (Technical report
SL-82-10)Cover title.
Final report.
"Prepared for Office, Chief of Engineers, U.S. Army
under Project 4AI61102AT22, Task BO, Work Unit 005."
Bibliography: p. 34.
1. Continuum mechanics. 2. Soil mechanics.
3. Soils--Testing. 4. Viscoelasticity. I. Rohani, ". -
Behzad. II. United States. Army. Corps of Engineers.
Baladi, George Y.
An elastic-viscoplastic constitutive model for : ... 1982.(Card 2)
Office of the Chief of Engineers. III. U.S. Army
Engineer Waterways Experiment Station. StructuresLaboratory. IV. Title V. Series: Technical report