INTRODUCTION TO CONTINUUM MECHANICS FOR ENGINEERS
CONTENTS 1 ONE-DIMENSIONAL CONTINUUM MECHANICS 11 KINEMATICS OF MOTION AND STRAIN 12 BALANCE OF MASS 13 BALANCE OF LINEAR MOMENTUM 14 BALANCE OF ENERGY 15 GENERAL BALANCE 16 THE ENTROPY INEQUALITY 17 EXAMPLE CONSTITUTIVE EQUATIONS 18 THERMODYNAMIC RESTRICTIONS 19 SMALL DEPARTURES FROM THERMODYNAMIC EQUILIBRIUM 110 SMALL DEPARTURES FROM STATIC EQUILIBRIUM 111 SOME FEATURES OF THE LINEAR MODEL BIBLIOGRAPHY 2 KINEMATICS OF MOTION 21 BODIES AND DEFORMATIONS 22 VELOCITY, ACCELERATION AND DEFORMATION GRADIENTS 23 TRANSFORMATION OF LINEAR, SURFACE AND VOLUME ELEMENTS 24 STRAIN KINEMATICS 25 INFINITESIMAL STRAIN KINEMATICS REFERENCES 3 EQUATIONS OF BALANCE 31 BALANCE OF MASS 32 BALANCE OF LINEAR MOMENTUM 33 BALANCE OF ANGULAR MOMENTUM 34 BALANCE OF ENERGY 35 THE ENTROPY INEQUALITY 36 JUMP EQUATIONS OF BALANCE - MATERIAL VERSIONS 4 MODELS OF MATERIAL BEHAVIOR 41 EXAMPLES 42 ISOTHERMAL ELASTICITY - THERMODYNAMIC RESTRICTIONS 43 ISOTHERMAL ELASTICITY - MATERIAL FRAME – INDIFFERENCE 44 ISOTHERMAL ELASTICITY - MATERIAL SYMMETRY 45 INCOMPRESSIBLE ISOTHERMAL ELASTICITY 46 THERMOELASTIC MATERIAL WITH HEAT CONDUCTION AND VISCOUS DISSIPATION-CONSTITUTIVE ASSUMPTIONS 47 THERMOELASTIC MATERIAL WITH HEAT CONDUCTION AND VISCOUS DISSIPATION-GENERAL THERMODYNAMIC RESTRICTIONS 48 THERMOELASTIC MATERIAL WITH HEAT CONDUCTION AND VISCOUS DISSIPATION-EQUILIBRIUM THERMODYNAMIC RESTRICTIONS 49 THERMOELASTIC MATERIAL WITH HEAT CONDUCTION AND VISCOUS DISSIPATION-MATERIAL FRAME – INDIFFERENCE 410 THERMOELASTIC MATERIAL WITH HEAT CONDUCTION AND VISCOUS DISSIPATION-MATERIAL SYMMETRY 411 CONSTITUTIVE EQUATIONS FOR A COMPRESSIBLE, CONDUCTING, VISCOUS FLUID 412 CONSTITUTIVE EQUATIONS FOR AN ISOTROPIC LINEAR THERMOELASTIC SOLID WITH HEAT CONDUCTION 5 MATERIALS WITH INTERNAL STATE VARIABLES 51 CONSTITUTIVE ASSUMPTIONS AND THERMODYNAMIC RESULTS 52 MAXWELL-CATTANEO HEAT CONDUCTOR 53 MAXWELLIAN MATERIALS 54 CLOSING REMARKS-ALTERNATE FORMS OF THE ENTROPY INEQUALITY ONE-DIMENSIONAL CONTINUUM MECHANICS IT IS OFTEN NOT CLEAR TO ENGINEERING STUDENTS THAT THERE IS A COMMON BASIS FOR THEIR COURSES IN THERMODYNAMICS, FLUID MECHANICS AND ELASTICITY THE PACE OF MOST UNDERGRADUATE CURRICULUMS IS SUCH THAT THERE IS NO OPPORTUNITY TO STRESS THE COMMON FEATURES OF THESE COURSES IN ADDITION, MANY UNDERGRADUATE ENGINEERING STUDENTS HAVE LIMITED SKILL WITH VECTOR ANALYSIS AND CARTESIAN TENSOR ANALYSIS THESE PROBLEMS MAKE IT AWKWARD TO TEACH A MODERN INTRODUCTORY COURSE IN CONTINUUM MECHANICS TO FIRST YEAR ENGINEERING GRADUATE STUDENTS EXPERIENCE HAS SHOWN THAT AN ELEMENTARY PREVIEW OF THE MODERN COURSE CAN BE A GREAT ASSET TO THE STUDENT THIS CHAPTER CONTAINS SUCH A PREVIEW IT IS A BRIEF SURVEY OF THE ELEMENTS OF CONTINUUM MECHANICS PRESENTED FOR ONE DIMENSIONAL CONTINUOUS BODIES THIS SURVEY ALLOWS THE STUDENT TO ENCOUNTER A NEW NOTATION AND SEVERAL NEW CONCEPTS WITHOUT THE PROBLEM OF LEARNING THREE DIMENSIONAL VECTOR AND TENSOR ANALYSIS THIS CHAPTER CONTAINS A DEVELOPMENT OF THE ONE DIMENSIONAL FORMS OF THE EQUATIONS OF BALANCE OF MASS, MOMENTUM AND ENERGY THE ENTROPY INEQUALITY IS PRESENTED, AND IT IS UTILIZED TO DERIVE THE THERMODYNAMIC RESTRICTIONS FOR A PARTICULAR MATERIAL MODEL 11 KINEMATICS OF MOTION AND STRAIN WE SHALL DENOTE BY U THE ONE DIMENSIONAL BODY THE SYMBOL X DENOTES AN ELEMENT OR PARTICLE OF THE BODY U IT IS USEFUL AT THIS POINT NOT TO DISTINGUISH BETWEEN THE BODY U AND THE PORTION OF ONE DIMENSIONAL SPACE IT OCCUPIES THUS, X IS A REAL NUMBER IT IS CUSTOMARY TO REFER TO X AS THE MATERIAL OR LAGRANGIAN COORDINATE OF THE PARTICLE THE SET OF MATERIAL COORDINATES IS A SUBSET OF THE REAL NUMBERS CALLED THE REFERENCE CONFIGURATION IF T DENOTES THE TIME AND E IS THE SET OF REAL NUMBERS, THEN THE DEFORMATION FUNCTION IS A FUNCTION ( , ) : T ? ? ? U E WHICH, FOR EACH T, MAPS U INTO ITS PRESENT CONFIGURATION WE WRITE ( , ) X X T ? = (111) WHERE X IS THE SPATIAL POSITION OR COORDINATE OF THE PARTICLE X AT THE TIME T THE SPATIAL COORDINATES ARE ALSO CALLED EULERIAN COORDINATES WE SHALL ASSUME THAT FOR EACH T , ? HAS AN INVERSE ? ? SUCH THAT 1( , ) X X T ? ? = (112) THESES ASSUMPTIONS INSURE THAT X AND X ARE IN ONE TO ONE CORRESPONDENCE FOR EACH T AND ARE, IN EFFECT, A STATEMENT OF PERMANENCE OF MATTER THE PARTICLE X CANNOT BREAK INTO TWO PARTICLES AS A RESULT OF THE DEFORMATION, AND TWO PARTICLES X AND X CANNOT OCCUPY THE SAME SPATIAL POSITION X AT THE SAME INSTANT OF TIME THE VELOCITY OF X AT TIME T IS CHAPTER ( , ) X T X T ? ? = ? (113) THE ACCELERATION OF X AT THE TIME T IS ( , ) X T X T ? ? = ? (114) THE DISPLACEMENT OF X AT THE TIME T IS ( , ) W X T X ? = ? (115) BECAUSE OF (112), WE CAN REGARD X , X AND W TO BE FUNCTIONS OF ( , ) X T OR ( , ) X T THE PAIR ( , ) X T ARE CALLED MATERIAL VARIABLES, AND THE PAIR ( , ) X T ARE CALLED SPATIAL VARIABLES CLEARLY, BY USE OF (111) AND (112) ANY FUNCTION OF ONE SET OF VARIABLES CAN BE CONVERTED TO A FUNCTION OF THE OTHER SET IF ? IS A FUNCTION OF ( , ) X T , THEN ITS MATERIAL DERIVATIVE, WRITTEN ? , IS DEFINED BY ( , ) X T T ? ? ? = ? (116) IF THE FUNCTION ? OF ( , ) X T IS DEFINED BY ( , ) ( ( , ), ) X T X T T ? ? ? ? = (117) AN ELEMENTARY APPLICATION OF THE CHAIN RULE YIELDS ( , ) ( , ) ( , ) X T X T X T X T T X ? ? ? ? ? ? ? = = + ? ? ? (118) EQUATION (118) GIVES THE MATERIAL DERIVATIVE IN TERMS OF SPATIAL DERIVATIVES FOR NOTATIONAL SIMPLICITY, WE SHALL WRITE (118) AS X T X ? ? ? ? ? = + ? ? (119) WHERE IT IS UNDERSTOOD THAT / T ? ? ? IS COMPUTED AT FIXED X , AND / X ? ? ? IS COMPUTED AT FIXED T AS AN ILLUSTRATION OF (119), WE CAN TAKE X ? = AND OBTAIN X X X X T X ? ? = + ? ? (1110) ONE-DIMENSIONAL CONTINUUM MECHANICS THE DEFORMATION GRADIENT IS DEFINED BY ( , ) X T F X ? ? = ? (1111) SINCE ? HAS AN INVERSE, IT IS TRIVIALLY TRUE THAT ( ( , ), ) X T T X ? ? ? = AND, THUS, ( , ) ( , ) X T X T X X ? ? ? ? ? = ? ? (1112) EQUATION (1112) SHOWS THAT F ? AND ( , ) X T F F X ? ? ? ? = = ? (1113) THE DISPLACEMENT GRADIENT IS DEFINED BY ( , ) W X T H X ? = ? (1114) IT FOLLOWS FROM (1111), (1114), AND (115) THAT H AND F ARE RELATED BY H F = ? (1115) IF DX DENOTES A DIFFERENTIAL ELEMENT OF THE BODY AT X , THEN IT FOLLOWS FROM (111) AND (1111) THAT DX FDX = (1116) EQUATION (1116) SHOWS THAT IF DX IS A MATERIAL ELEMENT AT X THEN FDX DX = IS THE DEFORMED ELEMENT AT X THEREFORE, F MEASURES THE LOCAL DEFORMATION OF MATERIAL IN THE NEIGHBORHOOD OF X IF X IS EXPRESSED AS A FUNCTION OF ( , ) X T , THEN X L X ? = ? (1117) IS THE VELOCITY GRADIENT BECAUSE ( , ) ( , ) X T X X X T X T X X X ? ? ?? ? ?? = = ? ? ? ? ? (1118) SHOW THAT F F L ? ? ? = ? (1120) NEXT WE SHALL STATE AND PROVE AN IMPORTANT RESULT KNOWN AS REYNOLD’S THEOREM OR THE TRANSPORT THEOREM CONSIDER A FIXED PORTION OF THE BODY IN X X X ? ? THE DEFORMATION FUNCTION, FOR EACH T , DEFORMS THIS PORTION INTO A REGION IN E WITHOUT LOSS OF GENERALITY, WE CAN TAKE THIS REGION TO BE X X X ? ? , WHERE ( , ) X X T ? = , ( , ) X X T ? = AND ( , ) X X T ? = REYNOLD’S THEOREM STATES THAT IF ? IS A SUFFICIENTLY SMOOTH FUNCTION OF X AND T , THEN ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) X X X X X T X T DX DX X T X X T X T X X T T ? ?? ? = + ? ?? ? ? ? I (1121) THE DERIVATION OF (1121) IS BEST APPROACHED BY UTILIZING A FORMALISM WHICH DOES NOT DEPEND UPON THE USE OF A DEFORMATION FUNCTION, AND ITS ASSOCIATED MOTION WE FIRST LET ( , ) X T ? BE THE INDEFINITE INTEGRAL, WITH RESPECT TO X , OF A FUNCTION ( , ) X T ? IN OTHER WORDS, LET ( , ) ( , ) CONSTANT X T X T DX ? = ? + ? (1122) IT FOLLOWS THEN THAT ( , ) ( , ) X T X T X ?? = ? ? (1123) AND, FOR TWO POINTS 1X AND 2X , ( , ) ( , ) ( , ) X X X T DX X T X T ? = ? ?? ? (1124) IF WE ALLOW THE LIMITS IN (1124) TO BE FUNCTIONS OF TIME, IE, ( ) ( ) ( , ) ( ( ), ) ( ( ), ) X T X T X T DX X T T X T T ? = ? ?? ? (1125) THE TOTAL DERIVATIVE OF ( ) ( ) ( , ) X T X T X T DX ? ? IS ONE-DIMENSIONAL CONTINUUM MECHANICS ( ( ), ) ( ( ), ) ( , ) ( ( ), ) ( ( ), ) ( ) ( ( ), ) ( ( ), ) ( ) X T X T D X T T D X T T D X T DX DT DT DT X T T X T T DX T T X DT X T T X T T DX T T X DT ? ? ? = ? ?? ?? = + ? ? ?? ?? ? ? ? ? ? (1126) IF (1125) IS USED, (1126) CAN BE WRITTEN ( ) ( ) ( ) ( ( ), ) ( ) ( ( ), ) ( ) ( , ) ( ( ), ) ( ( ), ) ( ) ( ) ( , ) ( ( ), ) ( ( ), ) X T X T X T X T X T T DX T X T T DX T D X T DX X T T X T T DT T DT T DT DX T DX T X T DX X T T X T T T DT DT ?? ?? ? = + ? ? ?? ? ? ? = ? + ? ?? ? ? ? (1127) THE NEXT FORMAL STEP IS TO INTERCHANGE THE ORDER OF INTEGRATION ON X WITH THE PARTIAL DIFFERENTIATION WITH RESPECT TO T THE RESULT OF THIS FINAL STEP IS SHOW THAT (1121) CAN BE DERIVED BY A CHANGE OF VARIABLES IN THE INTEGRAL ( , ) ( , ) ( , ) X X T X X T X T DX ? ? = = ? ? THE FIRST STEP IS TO WRITE ( , ) ( , ) ( , ) ( , ) ( ( , ), ) X X X T X X X X T X X T DX X T DX X T T FDX ? ? ? = = ? = ? = ? ? ? ? (1129) DIFFERENTIATE (1129) AND REARRANGE THE RESULT TO OBTAIN (1121) THE NAME "TRANSPORT THEOREM" IS SUGGESTED BY VIEWING THE LAST TWO TERMS IN (1121) AS THE NET TRANSPORTED OUT OF THE SPATIAL REGION X X X < < BY THE MOTION OF THE MATERIAL EQUATION (1128) AND ITS SPECIAL CASE (1121) ASSUME THAT ? IS DIFFERENTIABLE AND, THUS, FROM (1123), ? CONTINUOUS AS A FUNCTION OF X IN X X X < < NEXT WE WISH TO REMOVE THE ASSUMPTION THAT ? IS CONTINUOUS IN X X X < < AND DERIVE A GENERALIZATION OF (1121) THE DERIVATION SIMPLY UTILIZES THE GENERAL RESULT (1127) IN TWO DIFFERENT INTERVALS AND CAREFULLY JOINS THE RESULTS SO AS THEY APPLY TO THE INTERVAL X X X < < WE ASSUME ? IS CONTINUOUS EXCEPT AT A POINT ( ) Y T IN THE INTERVAL IF ? AND ? UNDERGO JUMP DISCONTINUITIES AT ( ) Y T SHOW THAT ? ? ? + ? ? 1X 2X ( ) Y T ONE-DIMENSIONAL CONTINUUM MECHANICS ? + ? + + + ?? = ? ? + ? ? = ? ? + ? ? = ? ? + ? ? + ? ? [ ] [ ] [ ] [ ] [ ] [ ][ ] [ ] [ ] AND ( ) ( ) IF ? UNDERGOES A JUMP DISCONTINUITY AT ( ) Y T , SHOW THAT ( , ) ( , ) X X DX X T X T X ?? = ? ?? + ? ? ? [ ] (1134) GIVEN A FUNCTION ? WHICH UNDERGOES A JUMP DISCONTINUITY AT ( ) Y T , REYNOLD’S THEOREM TAKES THE FORM ( , ) ( , ) ( , ) ( , ) ( , ) X X X X X T DX DX X T X X T T X T X X T Y ?? ? = + ? ? ?? + ? ? ? [ ] I (1135) WHERE ( ) Y T IS THE VELOCITY OF THE POINT ( ) Y T THE DERIVATION OF (1135) IS ELEMENTARY BECAUSE ? IS DIFFERENTIABLE IN ( ) X X Y T < < AND IN ( ) Y T X X < < , (1127) YIELDS ( , ) ( , ) ( , ) ( , ) Y Y X X D X T X T DX DX Y X T X X T DT T ? ?? ? = + ? ?? ? ? ? (1136) AND ( , ) ( , ) ( , ) ( , ) X X Y Y D X T X T DX DX X T X X T Y DT T ? ?? ? = + ? ?? ? ? ? (1137) THE ADDITION OF (1136) AND (1137) YIELDS (1135) IF (1134) IS USED, (1135) CAN ALSO BE WRITTEN ( , ) ( ) X X X X X X T DX DX X Y T X ?? ?? ? ? ? = + ?? ? ? ? ? ? ? ? ? ? [ ] I (1138) 12 BALANCE OF MASS IN THIS SECTION WE SHALL STATE THE ONE DIMENSIONAL FORM OF THE EQUATION OF BALANCE OF MASS THIS EQUATION IS THE FIRST OF FOUR FUNDAMENTAL PRINCIPLES WHICH FORM THE BASIS OF CONTINUUM MECHANICS THE OTHERS ARE BALANCE OF MOMENTUM, BALANCE OF ENERGY AND THE ENTROPY INEQUALITY THESE EQUATIONS OF BALANCE WILL BE DISCUSSED IN SUBSEQUENT SECTIONS WE SHALL DENOTE BY ? THE MASS DENSITY (MASS/LENGTH) OF U IN ITS DEFORMED CONFIGURATION THEREFORE, ( , ) X T ? ? = (121) THE CORRESPONDING QUANTITY IN THE REFERENCE CONFIGURATION IS ( ) R R X ? ? = (122) BALANCE OF MASS IS THE SIMPLE PHYSICAL STATEMENT THAT THE MASS OF THE BODY AND ANY OF ITS PARTS ARE UNALTERED DURING A DEFORMATION IF AN ARBITRARY PART OF THE BODY IS DEFINED BY X X X < < , THEN IT IS DEFORMED INTO X X X < < BY THE DEFORMATION, WHERE ( , ) X X T ? = AND ( , ) X X T ? = BALANCE OF MASS IS THE ASSERTION THAT X X R X X DX DX ? ? = ? ? (123) FOR ALL PARTS OF THE ONE DIMENSIONAL BODY U BECAUSE THE LEFT SIDE OF (123)IS INDEPENDENT OF T , AN ALTERNATE FORM OF BALANCE OF MASS IS X X DX ? = ? I (124) NEXT WE SHALL DERIVE THE LOCAL STATEMENT OF BALANCE OF MASS THE STATEMENT IS LOCAL IN THE SENSE THAT IT HOLDS AT AN ARBITRARY POINT X AT AN ARBITRARY TIME T RATHER THAN FOR AN INTERVAL X X X < < BY USE OF (1116), (123) CAN BE WRITTEN ( ) X R X F DX ? ? ? = ? (125) IF WE ASSUME THE INTEGRAND R F ? ? ? IS A CONTINUOUS FUNCTION OF X , THE FACT THAT (125) MUST HOLD FOR EVERY INTERVAL X X X < < FORCES THE FOLLOWING LOCAL STATEMENT OF BALANCE OF MASS: R F ? ? = (126) FOR REASONS THAT WILL BECOME CLEAR LATER, WE SHALL REFER TO (126) AS THE MATERIAL FORM OF THE LOCAL STATEMENT OF BALANCE OF MASS OTHER LOCAL STATEMENTS FOLLOW BY DIFFERENTIATION OF (126) IT FOLLOWS FROM (126) THAT F F F ? ? ? = + = I (127) ONE-DIMENSIONAL CONTINUUM MECHANICS IF WE NOW USE (1119) AND THE FACT THAT F ? , (127) YIELDS SHOW THAT (128) CAN BE WRITTEN IN THE ALTERNATE FORMS 1 L ? ? = I (129) AND X T X ? ? ? ? + = ? ? (1210) EQUATIONS (128), (129) AND (1210) ARE LOCAL STATEMENTS OF BALANCE OF MASS WHICH HOLD AT POINTS WHERE ? AND X ARE DIFFERENTIABLE IF THERE IS A POINT FOR WHICH ? AND X UNDERGO A JUMP DISCONTINUITY, WE MUST PROCEED FROM (123) MORE CAREFULLY IF ? AND X SUFFER A JUMP DISCONTINUITY AT ( ) Y T , IT FOLLOWS FROM (124) AND (1138) THAT X X X DX X Y T X ? ? ? ? ? ? ? ? + ? ? = ? ? ? ? ? ? ? [ ] (1211) EQUATION (1211) HOLDS FOR ALL ( , ) X X THE INTEGRAND IS ASSUMED TO BE CONTINUOUS AT ALL POINTS EXCEPT ( ) Y T IF ( , ) X X IS AN ARBITRARY INTERVAL WHICH DOES NOT CONTAIN ( ) Y T , THEN (1211) IMPLIES (1210) GIVEN (1210), (1211) THEN YIELDS ( ) USE (1210) AND SHOW THAT USE (1214) AND SHOW THAT ( , ) ( ) X X X X X T DX DX X Y ? ? ? ? = ? ? ? ? ? ? [ ] I (1215) 13 BALANCE OF LINEAR MOMENTUM IN THE THREE DIMENSIONAL THEORY, THE STATEMENT OF BALANCE OF MOMENTUM CONSISTS OF TWO PARTS THE FIRST IS THE STATEMENT CONCERNING THE BALANCE OF LINEAR MOMENTUM, AND THE SECOND IS A STATEMENT CONCERNING THE BALANCE OF ANGULAR MOMENTUM FOR A ONE DIMENSIONAL THEORY THE CONCEPT OF ANGULAR MOMENTUM DOES NOT ARISE THE LINEAR MOMENTUM OF THE PART OF U IN X X X < < IS X X XDX ? ? THE RATE OF CHANGE OF LINEAR MOMENTUM IS REQUIRED TO EQUAL TO THE RESULTANT FORCE ON THE PART OF U THE FORMAL STATEMENT IS WRITTEN X X XDX F ? = ? I (131) FOR ALL PARTS OF THE BODY U , WHERE F IS THE RESULTANT FORCE ACTING ON THE PART WE SHALL ASSUME THAT F CONSISTS OF TWO CONTRIBUTIONS AND WRITE ( , ) ( , ) X X F T X T T X T BDX ? = ? +? (132) THE QUANTITY ( , ) B X T IS CALLED THE BODY FORCE DENSITY (BODY FORCE/MASS), AND THE INTEGRAL OF B ? IS THE RESULTANT BODY FORCE ACTING ON THE PART OF U IN X X X < < THE QUANTITY ( , ) T X T IS A CONTACT FORCE IT RESULTS FROM THE CONTACT OF THE PART OF U IN X X X < < WITH THAT NOT IN X X X < < ( , ) T X T IS THE ONE DIMENSIONAL COUNTERPART OF STRESS IF ( , ) T X T > 0 (< 0) THE MATERIAL POINT IS IN TENSION (COMPRESSION) IF WE COMBINE (131) AND (132), WE OBTAIN ONE-DIMENSIONAL CONTINUUM MECHANICS ( , ) ( , ) X X X X XDX T X T T X T BDX ? ? ? = ? + ? ? (133) NEXT, WE WISH TO DEDUCE FROM (133) A LOCAL STATEMENT OF BALANCE OF LINEAR MOMENTUM FOR THE SAKE OF GENERALITY, WE ALLOW T , ? AND X TO SUFFER JUMP DISCONTINUITIES AT A POINT ( ) Y T IN U IT EASILY FOLLOWS FROM (1134) THAT ( , ) ( , ) X X T T X T T X T DX T X ? ? = ? ? ? [ ] (134) AND, FROM (1215), THAT ( ) X X X X XDX XDX X X Y ? ? ? = ? ? ? ? [ ] I (135) THESE RESULTS ALLOW (133) TO BE WRITTEN X X T X B DX X X Y T X ? ? ? ? ? ? ? ? ? ? + = ? ? ? ? ? ? [ ] [ ] (136) SINCE (136) MUST HOLD FOR ALL PARTS OF U , IT FOLLOWS BY THE SAME ARGUMENT THAT PRODUCED (1210) AND (1212) FROM (1211) THAT T X B X ? ? ? = + ? (137) FOR ALL ( ) X Y T ? , AND ( ) SHOW THAT AN ALTERNATE FORM OF THE ACCELERATION X IS SHOW THAT (138) CAN BE WRITTEN X Y T ? ? ? = [ ] (1310) AND 1/ T ? ? = [ ] [ ] (1312) EQUATION (1312) SHOWS THAT T [ ] AND 1/ ? [ ] MUST HAVE THE SAME SIGN EQUATION (1312) IS KNOWN AS THE RANKINE-HUGONIOT RELATION JUST AS (126) IS THE MATERIAL VERSION OF BALANCE OF MASS, EQUATION (137) CAN BE MANIPULATED INTO A MATERIAL VERSION OF BALANCE OF LINEAR MOMENTUM THIS MATERIAL VERSION IS R R T X B X ? ? ? = + ? (1313) EQUATION (1313) FOLLOWS BY MULTIPLICATION OF (137) BY F AND MAKING USE OF (126) 14 BALANCE OF ENERGY BALANCE OF ENERGY, OR THE FIRST LAW OF THERMODYNAMICS, IS THE STATEMENT THAT THE RATE OF CHANGE OF TOTAL ENERGY EQUALS THE RATE OF WORK OF THE APPLIED FORCES PLUS THE RATE OF HEAT ADDITION THE TOTAL ENERGY INCLUDES THE SUM OF THE INTERNAL ENERGY AND THE KINETIC ENERGY IF ( , ) X T ? IS THE INTERNAL ENERGY DENSITY (INTERNAL ENERGY/MASS), X X X DX ? ? + ? IS THE TOTAL ENERGY OF THE PART OF U IN X X X < < THE RATE OF WORK OR POWER OF THE APPLIED FORCES IS ( , ) ( , ) ( , ) ( , ) X X T X T X X T T X T X X T XBDX ? ? +? ONE-DIMENSIONAL CONTINUUM MECHANICS THE RATE OF HEAT ADDITION ARISES FROM HEAT GENERATED AT POINTS WITHIN THE BODY U AND FROM CONTACT OF THE PART OF U IN X X X < < IF ( , ) R X T DENOTES THE HEAT SUPPLY DENSITY (RATE OF HEAT ADDITION/MASS), THEN X X RDX ? ? IS THE RATE OF HEAT ADDITION RESULTING FROM HEAT GENERATED WITHIN THE BODY THE RATE OF HEAT ADDITION FROM CONTACT IS WRITTEN ( , ) ( , ) Q X T Q X T ? WHERE ( , ) Q X T IS THE HEAT FLUX THE MATHEMATICAL STATEMENT WHICH REFLECTS BALANCE OF ENERGY IS THEREFORE ( ) ( , ) ( , ) ( , ) ( , ) SHOW THAT (141) IMPLIES TX Q X XB R X X ? ? ? ? ? ? ? + = ? + + ? ? (142) FOR ALL ( ) X Y T ? AND ( )( ) SHOW THAT WHEN Q = [ ] (1212) AND (138) CAN BE USED TO WRITE (143) IN THE FORMS 1 ( ) T X Y ? ? ? + ? = [ ] (144) AND GIVEN THE DEFINITION (146), SHOW THAT (145) TAKES THE FORM DERIVE A MATERIAL VERSION OF (142) NEXT WE SHALL USE (137) TO DERIVE FROM (142) A THERMODYNAMIC ENERGY EQUATION IF (137) IS MULTIPLIED BY X , THE RESULT CAN BE WRITTEN T X X XB X ? ? ? ? = + ? (148) IF THIS EQUATION IS SUBTRACTED FROM (142), THE RESULT IS Q TL R X ?? ? ? = ? + ? (149) WHERE THE DEFINITION (1117) HAS BEEN USED THE TERM TX X ? ? IN (142) AROSE FROM THE RATE OF WORK OF THE CONTACT FORCES SINCE TX T X TL X X ? ? = + ? ? (1410) ONE-DIMENSIONAL CONTINUUM MECHANICS ON THE ASSUMPTION THAT NONE OF THE FIELD QUANTITIES UNDERGO JUMP DISCONTINUITIES, SHOW THAT ( , ) ( , ) X X X X X X DX Q X T Q X T RDX TLDX ?? ? = ? + + ? ? ? I (1411) AND X X X X X X X DX T X T X X T T X T X X T XBDX TLDX ? ? = ? + ? ? ? ? I (1412) EQUATIONS (1411) AND (1412) SHOW HOW THE STRESS POWER COUPLES THE INTERNAL ENERGY AND THE KINETIC ENERGY OF THE PART OF THE BODY U IN X X X < < 15 GENERAL BALANCE THE READER HAS PROBABLY NOTICED A FORMAL SIMILARITY BETWEEN THE THREE BALANCE EQUATIONS DISCUSSED THUS FAR IN THIS CHAPTER EACH BALANCE EQUATION IS A SPECIAL CASE OF THE FOLLOWING EQUATION OF GENERAL BALANCE: ( , ) ( , ) X X X X DX X T X T DX ?? ?? = ? ?? + ? ? I (151) IN EQUATION (151) THE LEFT SIDE REPRESENTS THE RATE OF CHANGE OF THE AMOUNT OF ? IN X X X < < THE TERM ( , ) ( , ) X T X T ? ?? REPRESENTS THE NET INFLUX OF ? , AND THE LAST TERM PRESENTS THE SUPPLY OF ? THE FOLLOWING TABLE SHOWS THE CHOICES OF ? , ? AND ? APPROPRIATE TO THE EQUATIONS OF BALANCE OF MASS, MOMENTUM AND ENERGY ? ? ? MASS MOMENTUM X T B ENERGY DERIVE THE FOLLOWING LOCAL STATEMENTS OF THE GENERAL BALANCE: X ?? ?? ?? = + ? (152) FOR ( ) X Y T ? , AND ( ) DERIVE A MATERIAL VERSION OF (152) 16 THE ENTROPY INEQUALITY THE ENTROPY INEQUALITY IS THE MATHEMATICAL STATEMENT OF THE SECOND LAW OF THERMODYNAMICS IN ORDER TO STATE THIS INEQUALITY, WE INTRODUCE THREE NEW QUANTITIES THE ENTROPY DENSITY (ENTROPY/MASS) IS DENOTED BY ( , ) X T ? THE ENTROPY FLUX IS DENOTED BY ( , ) H X T , AND THE ENTROPY SUPPLY DENSITY IS DENOTED BY ( , ) K X T THESE THREE QUANTITIES ARE REQUIRED TO OBEY THE FOLLOWING ENTROPY INEQUALITY OR CLAUSIUS-DUHEM INEQUALITY: ( , ) ( , ) X X X X DX H X T H X T KDX ?? ? ? ? + ? ? I (161) FOR ALL PARTS OF THE BODY U THE TEMPERATURE IS INTRODUCED BY FORCING THE RATIO OF ENTROPY FLUX TO HEAT FLUX TO EQUAL THE RATIO OF ENTROPY SUPPLY DENSITY TO HEAT SUPPLY DENSITY THE TEMPERATURE ( , ) X T ? IS DEFINED TO BE THE COMMON VALUE OF THESE TWO RATIOS, IE, ( , ) Q R X T H K ? = = (162) ONE-DIMENSIONAL CONTINUUM MECHANICS WE ALSO REQUIRE ? TO BE A POSITIVE NUMBER AND, THUS, ( , ) ( , ) ( , ) Q X T H X T X T ? = (163) AND ( , ) ( , ) ( , ) R X T K X T X T ? = (164) GIVEN (163) AND (164), (161) BECOMES ( , ) ( , ) ( , ) ( , ) X X X X Q X T Q X T R DX DX X T X T ? ?? ? ? ? ? ? + ? ? I (165) BY A NOW FAMILIAR ARGUMENT, (165) CAN BE WRITTEN / ( ) X X Q R Q DX X Y X ? ? ?? ?? ? ? ? ? ? + ? ? ? + ? ? ? ? ? ? ? [ ] (166) FOR ALL PARTS OF U THE FOLLOWING LOCAL INEQUALITIES ARE VALID: / Q R X ? ? ?? ? ? + ? ? ? (167) FOR ( ) X Y T ? , AND ( ) Q X Y ?? ? ? + ? [ ] (168) FOR ( ) X Y T = IF (167) IS WRITTEN Q Q R X X ? ?? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? (169) THE TERM Q R X ? ?? ? CAN BE ELIMINATED BY USE OF (149) THE RESULT OF THIS ELIMINATION IS ( ) Q TL X ? ? ?? ? ? ? ? + ? ? ? (1610) A MORE CONVENIENT VERSION OF (1610) RESULTS IF WE INTRODUCE THE HELMHOLTZ FREE ENERGY DENSITY DEFINED BY ? ? ?? = ? (1611) THIS DEFINITION ALLOWS (1610) TO BE WRITTEN ( ) ? IS A POSITIVE NUMBER, SHOW THAT ( ) ( ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) THE ENTHALPY DENSITY ? WAS DEFINED BY EQUATION (146) THE GIBBS FUNCTION IS DEFINED BY T ? ? ? = ? (1614) USE THESE DEFINITIONS AND SHOW THAT ( ) Q T X ? ? ?? ? ? ? ? ? ? ? ? (1615) AND ONE-DIMENSIONAL CONTINUUM MECHANICS SHOW THAT MATERIAL VERSIONS OF (167), (1612) AND (1617) ARE / R R R Q X ? ? ? ? ? ? + ? ? ? (1618) ( ) THROUGHOUT THIS CHAPTER, WE HAVE DEVELOPED JUMP EXPRESSIONS WHICH GOVERN BALANCE OF MASS, MOMENTUM AND ENERGY ACROSS A JUMP DISCONTINUITY WE HAVE ALSO, WITH (168), DEVELOPED A JUMP INEQUALITY WHICH FOLLOWS FROM THE ENTROPY INEQUALITY IT IS INTERESTING TO DEVELOP MATERIAL VERSIONS OF THESE JUMP EQUATIONS AS A FIRST STEP, COMBINE (126) AND (1212) AND SHOW THAT 1( ) F X Y ? ? = [ ] (1621) THE PHYSICAL QUANTITY 1 ( ) Y F Y X ± ± ? ± = ? IS THE VELOCITY IN THE REFERENCE CONFIGURATION OF THE IMAGE OF THE SPATIAL DISCONTINUITY THE NOTATION ± MEANS THAT THE EQUATION 1 ( ) Y F Y X ± ± ? ± = ? IS REALLY TWO EQUATIONS, ONE EVALUATED ON THE + SIDE OF THE DISCONTINUITY AND ONE EVALUATED ON THE ? SIDE IN ANY CASE, IT FOLLOWS FROM (1611) THAT CONSERVATION OF MASS FORCES Y = [ ] THUS, THE PHYSICAL QUANTITY Y IS ACTUALLY CONTINUOUS ACROSS THE DISCONTINUITY SHOW THAT MATERIAL VERSIONS OF EQUATIONS (138), (143) AND (168) ARE RY X T ? + = [ ] [ ] (1622) RY X TX Q ? ? + + ? = [ ] [ ] [ ] (1623) AND DERIVE THE MATERIAL VERSION OF THE RANKINE-HUGONIOT RELATION SHOW THAT MATERIAL VERSIONS OF (145) AND (147) ARE 1 ( ) R T T F ? ? ? + ? + = [ ] [ ] (1627) AND ( ) R F F T ? ? ? + + + = [ ] [ ] (1628) AS WITH (145) AND (146), THESE RESULTS ALSO ASSUME