2.2 Relation Between Mathematical & Engineering Constants Isotropic Materials Orthotropic Materials

Transcription

1 Chapter : lastic Constitutive quations of a Laminate.0 Introduction quations of Motion Symmetric of Stresses Tensorial and ngineering Strains Symmetry of Constitutive quations. Three-Dimensional Constitutive quations General Anisotropic Materials Orthotropic Materials Transversely Isotropic Materials Isotropic Materials. Relation Between Mathematical & ngineering Constants Isotropic Materials Orthotropic Materials. Constitutive quations for an Orthotropic Lamina Plane Strain Condition Plane Stress Condition.4 Constitutive quations for an Arbitrarily Oriented Lamina Coordinate Transformation Stress Transformation Strain Transformation Stiffness and Compliance Matri Transformation.5 ngineering Constants of a Laminate Lamina Laminate.6 Hygrothermal Coefficients of a Lamina.7 Summary

2 .0 INTRODUCTION u.0. quations of Motion of lastic Solids P (,, ) u quations of quilibrium (Kinetics) ρ ui, + fi i, j,, t ij j u u u quations of Kinematics (strain-displacement) ( ) ij u i, j + u j, i u u Constitutive quations (stress-strain) C i, j, k, l,, ij ijkl kl u u

3 .0. Symmetry of Stresses Consider a plane -.. A Tensorial and Contracted Notation Tensorial quilibrium in t t+ t t 0 in ( ) t ( ) t 0 Moment about A: t t 0 Similarly we can show, from - plane - plane Therefore, ij ji i, j,, Stress tensor is Symmetric. Contracted τ 4 or τ τ 5 or τ5 τ or τ 4 6 6

4 .0. Tensorial and ngineering Strains Tensorial Strains: ii ij i j + j i ( ) ij u i, j + u j, i u i, i i j normal strains. ( ) u, u, i j tensorial shear strain. ngineering shear strain ( ) γ ij ij + ji ui, j + uj, i Total shear strain. A ngineering Strains u u u u u γ u u γ u u γ u u u

5 Generalized Hooke s Law (-D Constitutive quation) Stress-Strain quation τ τ τ C is called the stiffness matri. Strain-Stress quation C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C i, j,,, 4,5,6 i ij j S i, j,,, 4,5,6 i ij j γ γ γ γ γ γ S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S τ τ τ S is called the compliance matri.

6 .0.4 Symmetry of Constitutive Matri Strain energy density, U0 i i () U C ij j i U0 i Cij j i U 0 Cij 0 j qn.() can be written as U0 j j i U C ji i j U0 j C 0 U 0 Since the order of differentiating a scalar quantity U 0 shouldnot change the result. Therefore, C ij C ji. Stiffness matri is symmetric. Similarly, S ij S ji i j j C ji ji i

7 . -D CONSTITUTIV QUATIONS (a) General Anisotropic Material (no plane of material symmetry). C C C C C C C C C C C C C C C C C C τ 4 C4 C4 C4 C44 C45 C γ 4 46 τ 5 C5 C5 C5 C54 C55 C γ 5 56 τ 6 C C C C C C γ Number of unknowns Because symmetry of C ij, number of unknowns 6 7/ (b) Specially Orthotropic Materials ( mutually perpendicular planes of material symmetry). Reference coordinate system is parallel to the material coordinate system. τ 4 τ 5 τ 6 Number of unknowns 9 C C C Sym C C C C C C 66 γ 4 γ 5 γ 6

8 Features No interaction between normal stresses (,, ) and shear strains (γ 4, γ 5, γ 6 ). Normal stresses acting along principal material directions produce only normal strains. No interaction between shear stresses (τ 4, τ 5, τ 6 ) and normal strains (,, ). Shear stresses acting on principal material planes produce only shear strains. No interaction between shear stresses and shear strains on different planes. That is shear stress acting on a principal plane produces a shear strain only on that plane.

9 (c) Transversely Isotropic Material An orthotropic material is called transversely isotropic when one of its principal plane is a plane of isotropy. At every point on this plane, the mechanical properties are the same in all directions. (-): Plane of Isotropy τ τ τ C C C C C C Sym C C C C 55 γ γ γ Number of unknowns 5

10 (d) Isotropic Material A material having infinite number of planes of material symmetry through a point. C C C Sym C C C τ C 44 τ C44 τ C C where C44 Number of unknowns Summary Material. Anisotropic material. Anisotropic elastic materials. Orthotropic material 4. Orthotropic material with transverse isotropy 5. Isotropic material C 44 γ γ γ Independent lastic constants

11 . Relations Between Mathematical and ngineering Constants (a) Isotropic Materials ( & ) / / / Definition: lastic Modulus () Stress/Strain / Poisson s Ratio () - Transverse strain/applied strain - /

12 / / / Normal Strains in in in Applied Stresses / / / / / / / / / Shear stresses Shear Strains γ 4 γ 5 Planes -, τ τ / G - τ τ / G - τ γ 6 τ / G

13 Constitutive quation i ij j S γ γ γ G G G τ τ τ { } [ ] {} S or { } [ ]{} C

14 Restrictions of lastic Constants Shear modulus G ( + ) for Shear modulus to be positive, > - Bulk modulus K ( ) for Bulk modulus to be positive, < / < < /

15 (b) Orthotropic Materials / / / Definition: lastic Modulus ( ) Stress/Strain / Poisson s Ratio ( ) - Transverse strain/applied strain - /

16 / / / Normal Strains Applied Stresses in in in / / / / / / / / / Shear Strains γ 4 γ 5 γ 6 Planes -, τ τ / G Shear stresses - τ τ / G - τ τ / G

17 Constitutive quation {} [ S]{ } γ γ γ G G G { } [ S] {} or { } [ C]{} from Symmetry of S- matri: Sij S S S S S S S ij ji Therefore or i j ij ji i j ji when i j That is, τ τ τ 4 5 6, and This is the well known Betti s reciprocal law of orthotropic material properties.

18 Stress-Strain quation { } [ C]{} Where [ ] [ ] C Coefficients of C are given by: C C C S C C C C G, C G, and C G Where

19 Transversely Isotropic Material ( Plane : ) G G G + ( ) Restrictions on lastic Constants of Orthotropic Materials From nergy Principles, Lempriere showed that the Strain nergy is Positive if the Stiffness and Compliance Matrices are Positive Definite. Mathematical Argument (a) If only one stress is applied at a time, then the work done is positive if and only when the corresponding direct strain is positive. That is when S ii > 0 Therefore:,,, G, G, and G > 0

20 (b) Under suitable constraints, it is possible to deform a body in one-direction. Then the work done will be positive if only when C ii > 0 C > 0 > 0 or < or < In general i ij < j Note all through was assumed to be greater than 0. This condition would give additional equations. (refer to R. M. Jones.)

21 . Constitutive quations of a Thin Orthotropic Lamina Two-Dimensional Bodies: Variation in stress and strain can be defined by two-coordinates. There are two types of problems. (a) Plane strain - Thick bodies z γ z γ yz 0 τ τ z yz 0, z, w, y, v (b) Plane Stress - Thin bodies z τz τyz 0 z γ z γ yz 0,, u

22 u,, y v,, z w,, Strain-Stress quation: γ S S S S S γ G Or Stress -Strain quation: γ Q Q Q Q Q γ G Or Where:

23 .4 Stress-Strain Relations for Arbitrary Orientation of a Lamina (a) Transformation of coordinates y - Material coordinate system - Reference coordinate system y P(,y) y Consider a point P(,y), its coordinates in system is Cosθ + ysinθ Sinθ + ycosθ or Cos θ Sinθ Sinθ Cos or θ y P(,y) y α α α y α y θ y Direction cosine matri α ij where I, and j,

24 (b) Stress Transformation We use tensors transfer stresses between the two coordinate systems ample: ij α α ij,, and kl, y, ij ik jl kl α α + α α + α α + α α y y yy If m Cosθ and n Sinθ Then m + mn + n y yy Similarly we can establish the other two stress components. Finally we can write m n mn n m mn mn mn m n { } [ T]{ } y or { } [ T ]{ } y where yy y [ T ] [ T] [ ] - is the stress transformation matri. T

25 [ T ] [ T] [ T( θ )] m n mn n m mn mn mn m n

26 (c) Strain Transformation { } [ T ]{ } yten Ten 0 0 { } [ ] yten T γ { } [ T ][ H]{ } yten ng { } [ H] [ T ][ H]{ } ng yng ng { } [ T ]{ } yng ng [ H] Where strain transformation matri is: [ T ] m n mn n m mn mn mn m n

27 (d) Stiffness Transformation Let { } [ Q] {} y y y in - y coordinate system Let us start with stress equation { } y [ ] { } T y [ T ] [ Q] {} y { } [ T ][ Q] [ T ] {} y y { } [ Q] {} {} [ T ] {} y { } [ T ][ Q] [ T ] {} y [ Q] [ T ][ Q] [ T ] y T T y

28 Q Q Q Q Q Q Q Q Q y s y yy y s sy ss Q Q T [ ] Q Q 0 0 Q T [ ] T where Q Q Q Q G 66

29 lements pf [Q] y matri Q m Q + n Q + m n Q + 4m n Q Qyy n Q + m Q + m n Q + 4m n Q ( ) Qy m n Q + m n Q + m + n Q 4m n Q ( ) + ( ) ( ) + ( ) Qs m nq mn Q + mn m n Q mn m n Q Qys mnq mnq + mn mn Q mn mn Q ( ) 66 Qss mnq + mnq mnq + m n Q Notice in the [ Q] y matri It is fully populated - means normal-shear coupling. Although 4 independent constants were used; we have 6 unknowns

30 (e) Compliance Matri {} [ T ]{} y [ ][ ] { } T S y y {} [ T ][ S] [ T ] { } {} [ S] { } y y y where [ S] [ T ][ S] [ T ] y T y γ y S S S S S S S S S y s y yy ys s sy ss yy y

31 S m S + n S + m n S + m n S Syy n S + m S + m n S + m n S ( ) Sy m n S + m n S + m + n S m n S ( ) + ( ) Ss m ns mn S + mn m n S mn m n S ( ) + ( ) Sys mns mns + mn mn S mn mn S ( ) 66 Sss 4m n S + 4m n S 8m n S + m n S 66 66

32 .5 ngineering Constants of an Arbitrarily Oriented Laminate Arbitrarily Oriented Lamina Let us eamine what happens when you apply in direction. We get... y y - in - in y y and shear strain, γ y γ y or θ y y y y y S S Shear coupling coefficient η s or γ η y s γ y η s S S y s η s y y S y η s S s

33 Shear Coupling Coefficients: S S S y s η s y S y S y η s S s η s ->Ratio of shear strain γ y to normal strain due to applied. η s ->Ratio of normal strain to shear strain γ y due to applied τ y. Similarly we have: η ys, η sy γ y y y η y G y η y G η η s ys y G s y sy y y yy y

34 ngineering Constants of an Arbitrarily Oriented Lamina y m m n n n m m n G ( ) + ( ) + ( ) + ( ) + n n m m m n m n G 4m n 4m n ( + ) + + G y ( ) + ( m n ) G y y m y m n n n m m n ( ) + ( ) + G ηs ηs mn mn mn m n m n n m G ( ) ( ) + G ( ) ( ) ηsy ηys mn mn mn mn n m m n G y ( ) ( ) + G

35 Variation of and y with Fiber Angle Material: 0 & G and Msi y Angle θ

36 Variation of Gy with Fiber Angle 0.8 G y Angle θ

37 y Variation of with Fiber Angle y Angle θ

38 Variation of ηs and ηys with Fiber Angle η ys η s Angle θ

39 ngineering Constants of a Laminate N-Layers ach Layer can have different Thickness, Orientation, and Material T N t i i Stress-Strain in ith Layer { } [ C] {} i i i Assumption: Strain is constant through out the laminate Average Stress in the laminate is: dz { } [ C] {} { } [ ] {} av T Cdz N [ ] {} T C i t i i { } [ C] {} or S av av For -D model stress-strain are si For -D model stress-strain are three av t {} [ ] { } av y av

40 ngineering Constants are: y z S Syy Szz G G G yz z y v S44 v S55 v S66 y z yz Sy S Sz S Szy Syy MmLamCode: mmtxlam: micromechanics and laminate analysis unidirectional code micro and laminate analysis of tetile fabric composite code

41 .6 Hygrothermal Coefficients of a Lamina.6. Coefficients of Thermal pansion (a) Isotropic Materials y Original b b l l panded due to T Coefficient of thermal epansion, α T αy T α T l l ' l T Units: in/in/ o F or m/m / o C

42 (B) Orthotropic Materials Deformed b b l l Original Coefficient of thermal epansion T In -direction α l l Thermal strains: l ' T ' T b b In -direction α b T T α T {} α T 0

43 .6. Coefficients of Moisture pansion All organic composites absorbs moisture. The absorption depends on the relative humidity to which it is eposed and its moisture content. For a given RH, temperature, and atmospheric pressure composite will have a saturation value. This is moisture content that the material will reach, if it is eposed for a very long time. This is a fied value for a material. The moisture content is epressed as percent change in weight of the material. Like thermal epansion,increase in moisture would also epands the material. The orthotropic materials have two coefficients of moisture epansion, one along the fiber and the other across the fiber. Deformed b b Change in moisture Μ l l Original Coefficient of moisture epansion T In -direction l l β l ' M b ' M In -direction β T b b Moisture strains: M { } β β M M 0 M

44 .6. Coefficients of Thermal & Moisture pansion for Lamina in Arbitrary Orientation y Recall the strain transformation: Thermal strains in -y due to T are: γ y y m n mn α T n m mn α T mn mn m n 0 Coefficients of thermal epansion in -y: T Where α α α T T { } [ T ]{ } y m n mn [ T ] n m mn mn mn m n mcosθ and nsinθ T y T y T α m α + n α α n α + m α α mn( α α ) T T T y T T T y T T T

45 Coefficients of thermal epansion in -y: T T T α m α + n α y T T T α n α + m α y T T T α mn( α α ) Coefficients of moisture epansion in -y: M M M β m β + n β y M M M β n β + m β β mn( β β ) y M M M

46 Summary.0 Introduction quations of Motion Symmetric of Stresses Tensorial and ngineering Strains Symmetry of Constitutive quations. Three-Dimensional Constitutive quations General Anisotropic Materials Orthotropic Materials Transversely Isotropic Materials Isotropic Materials. Relation Between Mathematical & ngineering Constants Isotropic Materials Orthotropic Materials. Constitutive quations for an Orthotropic Lamina Plane Strain Condition Plane Stress Condition.4 Constitutive quations for an Arbitrarily Oriented Lamina Coordinate Transformation Stress Transformation Strain Transformation Stiffness and Compliance Matri Transformation.5 ngineering Constants of a Laminate Lamina Laminate.6 Hygrothermal Coefficients of a Lamina