Chapter 1: Differential Form of Basic Equations

Transcription

1 MEG 74 Energ and Variational Methods in Mechanics I Brendan J. O Toole, Ph.D. Associate Professor of Mechanical Engineering Howard R. Hghes College of Engineering Universit of Nevada Las Vegas TBE B- (7) bj@me.nlv.ed Chapter : Differential Form of Basic Eqations 3-

2 Lectre 3 Otline Some Comments on Shear Strain Material Laws Eqilibrim Bondar Conditions Governing Eqations 3-

3 What Is the Difference Between Engineering Strain and Tensorial Strain Engineering and Tensorial Normal strains are defined the same: L/L Engineering Shear strain,, is defined as the change in angle of two perpendiclar aes as a material is deformed in shear. Tensorial shear strain is defined as ½ of engineering shear strain: ( )/ and ( )/ 3-3

4 Engineering Strain and Tensorial Strain It is sometimes convenient to write related eqations in matri form so that the are more compact. Albert Einstein devised a shorthand notation, Einstein s Indicial Notation, to be able to write eqations in an even more efficient manner (bt we will not be sing that here). All terms written with matri notation shold be eactl the same as if all the eqations were written ot eplicitl. Eample: Strain Energ Densit, W, is defined as the area nder the stress-strain crve. In the linear region of the crve, the area forms a triangle so: W ½ ( ) This eample assmes a -D problem with applied normal stresses in the - and - directions, and an applied shear stress. 3-4

5 Strain Energ Densit We wold like to be able to define strain energ densit in terms of matri operations: W ½ [][] Using engineering strain W W Engineering Strain and Tensorial Strain Using tensorial strain W ½ ( ) ( ) Tensorial Strain Mst Be Used In All Matri Operations ( ) This is the correct definition of W. Using engineering strain in matri eqations does not give the correct W. Using tensorial strain in matri eqations does give the correct W. 3-5

6 Material Laws (Constittive Relations) (Stress - Strain Eqations) The stress-strain eqations or constittive relations are a good eample of the seflness of the indicial notation. The most basic constittive relation that we first learn in Mechanics of Materials is the -D Hookes Law eqation: E, Stress eqals Yong's modls times strain. Later, we learned that if there is strain in more than one direction, the stress will be a fnction of all strains. 3-6

7 General 3-D Constittive Relations The first two (of nine) stress eqations can be written as shown below. Each of the nine stress components is a fnction of all nine strain components. Each Q variable is a different fnction of material properties. Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 3-7

8 General 3-D Constittive Relations Both the shorthand and fll matri versions of the 3-D constittive eqations are shown below. Q kl kl where i, j, k and l represent, and. Sometimes the nmbers, and 3 are sbstitted for, and. Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 3-8

9 Simplifing the Constittive Relations The general constittive eqations have 8 elastic constants. Lckil, the nmber of constants is redced for most practical materials. Both the stress and strain tensors are smmetric. ji and ji This redces the nmber of independent stresses and strains to

10 Constittive Relations Since there are onl si independent stresses and strains, another shorthand notation is introdced sing nmeric sbscripts. Let and and 3 and 3 4 and 4 5 and 5 6 and 6 3-

11 Constittive Relations This redces the nmber of independent material constants to Q Q Q3 Q4 Q5 Q6 Q Q Q3 Q4 Q5 Q6 Q3 Q3 Q33 Q43 Q53 Q63 Q4 Q4 Q34 Q44 Q54 Q64 Q5 Q5 Q35 Q45 Q55 Q65 Q6 Q6 Q36 Q46 Q56 Q66 A thermodnamics proof can be sed to show that the Q matri itself is smmetric, or Q Q ji. This redces the nmber of independent constants to

12 Smmetr of Stiffness Matri Smmetr of the stiffness matri is shown b considering the definition of Strain Energ: W The se of Einstein s Indicial Notation simplifies the proof. The proof is shown here bt we will not be covering indicial notation in this corse. Q W W Q kl kl kl Q Q kl These eqations can be rewritten with different sbscripts : kl Q kl This implies that Q, the stiffness matri is smmetric and redces the nmber of independent constants from 36 to. kl kl Q kl kl kl Q Both epressions for the strain energ are valid and can be eqated : Q Q Therefore : kl kl kl kl kl 3-

13 Anisotrop and Material Smmetr General Anisotrop A material with different material properties in all directions ehibits general anisotrop. independent elastic constants are reqired to define the stress-strain relationship for this tpe of material. Orthotropic Material Has 3 mtall orthogonal planes of elastic smmetr e.g.: A material with the same properties in the and - directions has elastic smmetr abot the - plane. Most composite materials ehibit elastic smmetr abot three planes Reqires 9 independent elastic constants 3-3

14 3-4 Orthotropic Stress-Strain Eqations or i Q j, where Q is know as the stiffness matri. Inverting the matri eqation ields: S S S S S S S S S S S S Q Q Q Q Q Q Q Q Q Q Q Q or i S j, where S is know as the compliance matri. [S] [Q] -

15 Compliance Matri The compliance matri vales are easier to define than the stiffness matri vales S S S3 S S S3 S3 S3 S33 S44 S55 S S E S 44 G 3 S E S 55 G 3 S 33 E 33 S 66 G S S E E S 3 S 3 3 E 3 E 33 S 3 S 3 3 E 3 E

16 Stiffness Matri TheQ components are fond b inverting S Q E ( 3 3 ) Q 44 G 3 Q E ( 3 3 ) Q 55 G 3 Q 33 E 33 ( ) Q 66 G Q E ( 3 3 ) E ( 3 3 ) Q 3 E ( 3 3 ) E 33 ( 3 3 ) Q 3 E ( 3 3 ) E 33 ( 3 3 )

17 3-7 Isotropic Stress-Strain Relations ( ) ( ) ( ) E ( )( ) E

18 Isotropic Stress-Strain Relations (Matri Form of Eqations) E E (Inde Form of Eqations) kk ( )( ) ( ) E λδ or δ kk µ Where λ and µ are known as Lame s constants. Note that onl independent constants are needed to describe isotropic material behavior. E Also note that µ G shear stiffness or modls of rigidit 3-8

19 Common -D Conditions: Plane Stress Applied to thin flat plates where the loads are generall in the plane of the plate. Assme normal and shear stresses in the -direction are ero: Assme all other stresses and strains do not var throgh the thickness. Sbstitte these assmptions into the general isotropic material law eqations to get the following: and ( ) ( ) 3-9

20 Common -D Conditions: Plane Strain Applied to long strctres where loads are in the transverse direction (long pressried clinders). Can be applied to other strctres where strain is restricted in the -direction. Assme strains along the long ais of the clinder are ero -direction): Sbstitte these assmptions into the general isotropic material law eqations to get the following: ( ) and 3-

21 3- Material Laws for Plane Problems Plane Stress: E Plane Strain: ( )( ) E

22 3- Stress Resltants n n n t It is sometimes convenient to define stress resltants (n, n, n ) as an alternative for stresses. Calclate the stress resltant as force/width (not force/thickness). Units are force/length. / / / / / / t t t t t t d n d n d n ( )( ) Et n n n Plane Strain Plane Stress Et n n n

23 Total State of Strain Thermal strains or initial strains can be added to strains cased b applied loads: E ( o ) E A change in temperatre cases the following strains in an isotropic bod: o o o o o o α T o where α is the coefficient of thermal epansion. 3-3

24 D Eqilibrim Element State of stress acting on a differential planar element at point O in a -D bod. d d Volmetric (or bod) forces are represented b p v F d p V o d p V d d The volmetric differential element has dimensions of d, d, and d into the page. The -eqilibrim eqation is shown below: ( ) ( ) d dd d dd dd dd p ddd : V Dividing b ddd ields: p V 3-4

25 3-5 D Eqilibrim Eqations Writing a force balance eqation in the -direction and a moment balance eqation abot point O ields: V p Writing the -D eqilibrim eqations in matri form: V V p p p D V T

26 Stress Fnctions It is often convenient to epress the different stresses in terms of a single stress fnction. Then, instead of solving for 3 different stresses, there will onl be one nknown stress fnction. Air s stress fnction (ψ) is most common. In the absence of bod forces it is: ψ ψ ψ The stress fnction mst satisf the eqilibrim eqations. Verif that these eqations satisf eqilibrim. 3-6

27 Bondar Conditions Force Bondar Conditions along this edge. Displacement Bondar Conditions along this edge. S p Strctral Bod S The entire bondar of the strctre mst be defined as S OR S p. on S Force bondar condition ma be pressre, moment, point load, or ero load. Where are the prescribed displacements. Applied srface forces (per nit area) are referred to as srface tractions p. p p on S p 3-7

28 3-8 Governing Eqations ( ) i k k i ik kk Vi i P,, µ λδ Eqilibrim (3): Material Law (6): Strain Displacement Eqations (6): 5 Eqations with 5 nknowns:

29 3-9 Displacement Formlation of Governing Eqations Matri Form of Eqations D Strain Displacement Eqations: Material Law: Eqilibrim: () () (3) E D T p V Sbstitte () into () into (3): D T E D p V G p G p G p V V V Now there are onl 3 eqations and 3 nknowns:,,,

30 Displacement Formlation of Governing Eqations Shorthand Notation G i ( λ G) p k, ki Vi If there are no bod forces than this can be simplified even frther: i This is the classic biharmonic differential eqation that appears freqentl in mathematics. is the Laplacian or harmonic operator: i 3-3

31 Net Class Stress Analsis Engineering Beam Theor Torsion Theor 3-3