ME 7502 Lecture 2 Effective Properties of Particulate and Unidirectional Composites
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1 ME 75 Lecture Effective Properties of Particulate and Unidirectional Composites Concepts from Elasticit Theor Statistical Homogeneit, Representative Volume Element, Composite Material Effective Stress- Strain Relations Particulate composite effective moduli Unidirectional composite effective moduli Lamina failure theories Lamina constitutive relations ME 75 Lecture
2 CONCEPTS FROM ELASTICITY THEORY STRESS AND EQUILIBRIUM SURFACE TRACTION AND STRESS STRESS EQUILIBRIUM DISPLACEMENT AND STRAIN NORMAL AND SHEAR STRAIN, TENSOR STRAIN STRAIN COMPATIBILITY LINEAR ELASTIC STRESS-STRAIN- TEMPERATURE RELATIONS ISOTROPIC MATERIALS ORTHOTROPIC MATERIALS TRANSVERSELY-ISOTROPIC MATERIALS ME 75 Lecture
3 STRESS AND EQUILIBRIUM DEFINITION OF STRESS SURFACE TRACTION INTERNAL DISTRIBUTED FORCE ON THE CUT SURFACE OF A STRUCTURE ME 75 Lecture 3
4 SURFACE TRACTION VECTOR ON A PLANE AT A POINT A - Infinitesimal area T not perpendicular to surface, not parallel to n T F lim A Ti T j T A z k T - EXPRESSED IN FIXED GLOBAL COORDINATES X, Y, Z ME 75 Lecture 4
5 TRACTION AS SURFACE STRESS NORMAL AND PARALLEL TO THE SURFACE where, n T n n n i n j n z k n i cos( n, i) i,, z since T n t t t T n T n t Therefore, the tangential unit vector t is n T nn t t ME 75 Lecture 5
6 STRESS AT A POINT POINT IS SURROUNDED BY A BOX WHOSE SIDES ARE PARALLEL TO COORDINATE AXES X FACE: OUTWARD NORMAL IS IN X DIRECTION -X FACE: OUTWARD NORMAL IS IN X DIRECTION ETC. ME 75 Lecture 6
7 SURFACE TRACTION COMPONENTS ON RECTANGULAR VOLUME ELEMENT SURROUNDING POINT DEFINE STRESS FROM FORCE EQUILIBRIUM:, ETC. z z z z FROM MOMENT EQUILIBRIUM: z z AND, SINCE, ETC., ij ji INDEPENDENT STRESS COMPONENTS REDUCE TO 6 STRESS TENSOR IS SYMMETRIC ME 75 Lecture 7
8 COMPONENTS OF STRESS AT A POINT IN THREE- DIMENSIONAL RECTANGULAR CARTESIAN COORDINATES: NOTE THAT ij ij (,, z) NORMAL STRESS COMPONENTS ALWAYS OCCUR ON AN ELEMENT IN PAIRS SHEAR STRESS COMPONENTS OCCUR IN QUADRUPLES IF THE MAGNITUDE AND SENSE OF A STRESS COMPONENT ON ONE FACE IS KNOWN, THE SAME STRESS COMPONENT WILL ALSO BE KNOWN ON OTHER FACES ON WHICH IT ACTS. ME 75 Lecture 8
9 MATRIX REPRESENTATION OF STRESS: [ ] z z z z zz z z z z zz τ τ z τ τ z τ z τ z zz SYMMETRY DIFFERENT NOTATION SIGN CONVENTION FOR STRESS COMPONENTS: FACE OUTWARD DIRECTION OF STRESS SIGN OF STRESS NORMAL DIRECTION COMPONENT COMPONENTS POSITIVE () POSITIVE () POSITIVE () NEGATIVE (-) NEGATIVE (-) POSITIVE () NEGATIVE (-) POSITIVE () NEGATIVE (-) POSITIVE () NEGATIVE (-) NEGATIVE (-) ME 75 Lecture 9
10 DIFFERENTAL EQUATIONS OF STRESS EQUILIBRIUM ME 75 Lecture
11 ME 75 Lecture IF NO BODY FORCES, GET j ji OR j ij DIFFERENTAL EQUATIONS OF STRESS EQUILIBRIUM
12 SURFACE ELEMENT EQUILIBRIUM ME 75 Lecture
13 DISPLACEMENT AND STRAIN DEFINITION OF DISPLACEMENT DISPLACEMENT u u DISPLACEMENT VECTOR u -component of displacement ui vj wk v -component of displacement w z-component of displacement ME 75 Lecture 3
14 STRAIN & STRAIN- DISPLACEMENT EQUATIONS ME 75 Lecture 4
15 STRAIN-DISPLACEMENT EQUATIONS (cont d) ME 75 Lecture 5
16 STRAIN-DISPLACEMENT EQUATIONS (cont d) ME 75 Lecture 6
17 STRAIN COMPATIBILITY DISPLACEMENTS MUST BE SINGLE-VALUED & CONTINUOUS (NO HOLES OPEN IN THE STRUCTURE; PARTICLES OF MATTER CANNOT OCCUPY SAME SPACE AT SAME TIME) STRAINS MUST BE DERIVABLE FROM SINGLE-VALUED, CONTINUOUS DISPLACEMENTS (COMPATIBLE) CONSIDERING EQUATIONS (3) TO (5), IT IS POSSIBLE TO HAVE VALUES OF,, AND γ SUCH THAT WHEN (3) TO (5) ARE INTEGRATED TO OBTAIN u(,) and v(,), A UNIQUE SOLUTION CANNOT BE OBTAINED. THEREFORE, THERE MUST BE SOME RELATIONSHIP BETWEEN,, AND γ. CONSIDERING (5): IN TENSOR FORM, THE COMPATIBILITY EQUATIONS ARE: EQUATION (9) REPRESENTS A TOTAL OF 8 (3 4 ) EQUATIONS; HOWEVER, ONLY SIX ARE INDEPENDENT. ME 75 Lecture 7
18 ME 75 Lecture 8 6 INDEPENDENT STRAIN COMPATIBILITY EQUATIONS: COMPATIBILITY EQUATIONS ARE EQUIVALENT TO STRAIN- DISPLACEMENT EQUATIONS, AND GIVE NO NEW INFORMATION z z z z z z z z z z z z zz z z zz z z z z zz z z z z γ γ γ
19 LINEAR ELASTIC STRESS-STRAIN RELATIONS ELASTIC MODULI temp incr T Coefficient of linear thermal epansion α Α Α / T Α α Α T ME 75 Lecture 9
20 IMPORTANT POINTS FOR ISOTROPIC OR ORTHOTROPIC MATERIALS (PRINCIPAL MATERIAL AXES). UNIAXIAL NORMAL STRESS A CAUSES NORMAL STRAINS. FOR EXAMPLE A A /E A AND T - AT A - AT A /E A. PURE SHEAR STRESS τ AT CAUSES ONLY SHEAR STRAIN γ AT (NO OTHER SHEAR STRESSES, NO OTHER SHEAR STRAINS) 3. UNIAXIAL NORMAL STRESS CAUSES ONLY NORMAL STRAINS (NO SHEAR STRAINS) 4. TEMPERATURE CHANGE T CAUSES ONLY NORMAL STRAINS (NO SHEAR STRAINS) ME 75 Lecture
21 γ γ γ zz z z ISOTROPIC MATERIALS (CAST MATERIALS, MOST METALS) E E zz E τ z G τ z G τ G PROPERTIES ARE DIRECTION-INDEPENDENT ONLY -E, -G, -, -α E E E E E E τ z, z G τ z, z G τ, G zz zz α T α T α T OR ij ij E E NOTE: CAN PROVE THAT (NO RELATION BETWEEN E AND ALONE) ME 75 Lecture kk δ 3 INDEPENDENT THERMOELASTIC CONSTANTS ij α Tδ E G( ) { E, OR E, G OR, G OR K,G} AND α E 9KG /(3K G) ij
22 ME 75 Lecture INVERTING PREVIOUS EQUATIONS YIELDS z z z z z z z z zz zz zz zz zz G G G G G G T E T E T E τ γ τ τ γ τ τ γ τ α α α,,, ) ( ) ( ) ( OR ij ij kk ij ij T E δ α δ
23 ORTHOTROPIC MATERIAL RELATIONS ME 75 Lecture 3
24 ORTHOTROPIC MATERIAL RELATIONS ME 75 Lecture 4
25 ME 75 Lecture 5 TRANSVERSELY ISOTROPIC MATERIALS L α T α α α α 3, T E E E T E E E T E E T T T TT L LT T T TT T L LT L T TL L α α α ) ( L L L L T T G G G G G G,,, τ τ γ τ τ γ τ τ γ
26 SINCE T.I. PROPERTIES ARE NOT DIRECTION-DEPENDENT, ET G T ( T ) ALSO, AS WITH GENERAL ORTHO, E LT L E TL T 7 INDEPENDENT THERMOELASTIC CONSTANTS: E S, or G S, or s, α s AN APPROXIMATION: MOST TRANSVERSELY-ISOTROPIC COMPOSITES HAVE G L ~G T ME 75 Lecture 6
27 Anisotropic material properties: calculation using phase (fiber or particulate) and matri properties ME 75 Lecture 7
28 EFFECTIVE COMPOSITE PROPERTIES Statistical Homogeneit: to calculate effective properties, it is first necessar to introduce a representative volume element (RVE), which must be large compared to tpical phase region dimensions (i.e., reinforcement diameters and spacing) RVE must be large enough so that average stress in RVE is unchanged as size increases: ME 75 Lecture 8
29 EFFECTIVE COMPOSITE PROPERTIES Effective properties of a composite material define relations between averages of field variables and C ijkl* and S ijkl* are reciprocals of one another Overbars denote RVE averages ME 75 Lecture 9
30 PARTICULATE REINFORCED COMPOSITE MODULI Provided dispersion of particulate reinforcement is uniform, and provided orientation of non-spherical particulates is random, stress-strain relations of such composite materials will be effectivel isotropic Two independent elastic moduli For convenience, these are selected to be the bulk modulus (K) and the shear modulus (G) All other elastic constants can be defined in terms of K and G ME 75 Lecture 3
31 PARTICULATE REINFORCED COMPOSITE MODULI Effective elastic constants of particulate reinforced composites are obtained using multi-phase material solutions from elasticit theor Eact solutions are possible onl in the case of spherical particles Approimate (bounding theor) results are used for other cases, such as non-spherical particles Eample of a lower bound result is Arbitrar Phase Geometr (APG) lower bound on G* G* G m G i G m v 6v 5G i m m ( K m (3K m G ME 75 Lecture 3 m 4G ) m )
32 RESULTS OF BOUNDING THEOREMS FOR PARTICULATE REINFORCED COMPOSITE MATLS Best lower bound is from Arbitrar Phase Geometr (APG) bound Best upper bound is from Composite Spheres Assemblage (CSA) bound ME 75 Lecture 3
33 UNIDIRECTIONAL COMPOSITE PROPERTIES We are first interested in finding effective properties of a unidirectional composite (pl) Unidirectional composite is transversel isotropic 3 plane is plane of isotrop ME 75 Lecture 33
34 UNIDIRECTIONAL COMPOSITE PROPERTIES Material stiffness matri for a transversel isotropic material is There are 5 independent elastic constants and independent CTE s ME 75 Lecture 34
35 UNIDIRECTIONAL COMPOSITE PROPERTIES The relations of a transversel isotropic material are frequentl written as ME 75 Lecture 35
36 UNIDIRECTIONAL COMPOSITE PROPERTIES For the unidirectional composite subjected to a homogeneous stress state (e.g., uniform tension), states of stress and strain are not uniform but highl comple Variations of stress and strain on an transverse plane are random No reason that one plane should be different from another, other than statistical variation Such a material is statisticall homogeneous, which allows the definition of effective elastic properties to be specified b relations between average stress and average strain Unidirectional composite is statisticall transversel isotropic due to the random placement of fibers in the pl or lamina ME 75 Lecture 36
37 UNIDIRECTIONAL COMPOSITE PROPERTIES Effective relations of a lamina: ME 75 Lecture 37
38 COMPOSITE CYLINDERS ASSEMBLAGE MODEL Micromechanical methods are used to compute the effective elastic properties of a unidirectional composite material in terms of fiber and matri properties and these phase volume fractions Composite Clinders Assemblage (CCA) model is used for aial properties Generalized Self-Consistent Scheme (GSCS) is used to calculate transverse properties CCA model GSCS model ME 75 Lecture 38
39 PHILOSOPHY OF CCA MODEL Unidirectional composite is considered as an assemblage of composite clinders Varing diameters but constant d f / d m ratio Properties of homogeneous clinder which can be used to replace each heterogeneous clinder provide the effective lamina properties 5 elastic, CTE s Schematic of CCA model ME 75 Lecture 39
40 CONCEPT OF GENERALIZED SELF- CONSISTENT SCHEME (GSCS) Single composite clinder is embedded in the model effective material Properties of effective material which satisf continuit of displacements and tractions across interface between matri (a<r<b) and effective material are sought Motivation for GSCS: CCA can onl bound transverse properties of pl Schematic of GSCS model ME 75 Lecture 4
41 UNIDIRECTIONAL COMPOSITE PROPERTIES Effective Elastic Properties are obtained b solving closed form solutions to Theor of Elasticit problems Uniform stress states are applied to the two (fiber and matri) phase composite clinder in clindrical coordinates Details are provided in tetbooks b Robert Jones, R.M. Christensen and others ME 75 Lecture 4
42 COMPLETE RESULTS OF CCA ANALYSES fiber; matri ME 75 Lecture 4
43 RESULTS OF CCA ANALYSES (cont d) fiber; matri ME 75 Lecture 43
44 Composite material strength and failure theories ME 75 Lecture 44
45 COMPOSITE STRENGTH & FAILURE THEORIES Unidirectionall reinforced composites are tpicall emploed as lamina within laminates and thus are subjected to combined stress states Stress state is at least plane stress Therefore it is usuall necessar to consider strength of lamina under combined stress state Best approach is to consider analtical theor with limited number of eperiments ME 75 Lecture 45
46 PRACTICAL LAMINA FAILURE CRITERION Fundamental assumption is that for plane stress state there eists a failure criteria of the form F(,, ) Schematic representations are: ME 75 Lecture 46
47 LAMINA FAILURE CRITERION (cont d) Stress states inside curves (F<) do not induce lamina failure Stress states on or outside curves (F or F>) induce lamina failure ME 75 Lecture 47
48 LAMINA FAILURE CRITERION (cont d) Two common tpes of failure criteria Maimum Stress / Strain Quadratic Interaction Maimum Stress or Strain Assumption is that failure occurs when an single stress (or strain) component reaches its ultimate value, regardless of values of other components Advantages: Ver simple to use and allows failure modes to be identified Disadvantages: Not necessaril realistic since it ignores stress interaction It is sometimes not conservative, depending on stress state ME 75 Lecture 48
49 LAMINA FAILURE CRITERION (cont d) Quadratic Interaction Failure Criterion General form is given b the equation F F F 66 F F F () Three popular forms of quadratic interaction: ) Tsai-Hill: () X aial strength (tensile or compressive) Y transverse strength (tensile or compressive) S shear strength ME 75 Lecture 49
50 LAMINA FAILURE CRITERION (cont d) Quadratic Interaction Failure Criterion ) Tsai-Wu Failure Criterion uses F F F 66 F F F () (3) ME 75 Lecture Requires bi-aial testing Tsai-Wu is basicall a curve fit and does not specif failure mode 5
51 LAMINA FAILURE CRITERION (cont d) (4) ME 75 Lecture 5
52 LAMINA FAILURE CRITERION (cont d) Quadratic Interaction Failure Criterion 3. Hashin Failure Criteria ME 75 Lecture 5
53 LAMINA FAILURE CRITERION (cont d) Hashin Failure Criterion (cont d) Convenient method for eamining failure criteria eperimentall is b means of off-ais specimens; here stress state is given b Testing in tension shows that for small off-ais angles, failure is in the fiber mode; for large angles, failure is b matri mode Angle θ o separating the two modes is found when (5) and (6) are satisfied simultaneousl in terms of (7); result is ME 75 Lecture 53
54 LAMINA FAILURE CRITERION (cont d) Off-ais test specimen Hashin Criteria Failure Modes ME 75 Lecture 54
55 LAMINA FAILURE CRITERION (cont d) Performance of Hashin Interaction Failure Criteria (for Gl/Ep composites) ME 75 Lecture 55
56 LAMINA CONSTITUTIVE RELATIONS PLANE STRESS USUALLY USE LAMINATED COMPOSITES IN PLATE OR SHELL FORM WHERE PRIMARY STRESSES ARE IN THE PLANE OF THE LAMINA (-) (PLANE STRESS). LAYER STRESS-STRAIN BEHAVIOR THEN BECOMES: E [ ] α T OR DEFINE [Q t ] MATRIX: OR E G E E Q Q [ ] E E Q Q Q 66 α T G α α T α α T t t { } [ Q ]{ { } { α } T} (D-) ME 75 Lecture 56
57 ME 75 Lecture 57 { } [ ]{ } { } T S t t t α THE INVERSE STRAIN-STRESS RELATIONSHIP IS GIVEN BY T G E E E E α α T S S S S S 66 α α (D-3) LAMINA CONSTITUTIVE RELATIONS (cont d)
58 LAMINA CONSTITUTIVE RELATIONS (cont d) CONSTITUTIVE RELATIONS IN OFF-AXIS (X,Y) LAMINATE COORDINATES HAVE WANT t t { } [ Q ]{ { } { α } T} t t t { } [ Q ]{ { } { α } T} θ a. TRANSFORMATION OF STRESS AND STRAIN STRESS τ τ θ ME 75 Lecture 58
59 ME 75 Lecture 59 ENGINEERING STRAIN TENSOR STRAIN ) ( n m mn mn mn m n mn n m γ γ γ γ ) ( n m mn mn mn m n mn n m STRESS ) ( ) sin (cos cos )sin ( cos sin cos sin cos sin sin cos n m mn mn mn m n mn n m τ θ θ τ θ θ τ τ θ θ τ θ θ τ θ θ τ θ θ θ sinθ, cos m n LAMINA CONSTITUTIVE RELATIONS (cont d)
60 LAMINA CONSTITUTIVE RELATIONS (cont d) DEFINE TENSOR -D TRANSFORMATION MATRIX THEN [ ] T CL m n mn { } [ T ]{ } CL n m mn mn mn m n m cos θ, n sinθ (D-4) (D-5a, b) { t} { t} T [ T ]{ t} { t α α } T} CL NOTE: [ ] T CL m n mn n m mn mn m mn n (D-6) THIS IS USED FOR STRESS AND TENSOR THERMOMECHANICAL STRAIN (WOULD NEED DIFFERENT [ ] e T CL IF ENGINEERING STRAIN!) ME 75 Lecture 6
61 LAMINA CONSTITUTIVE RELATIONS (cont d) b. TRANSFORMED LAMINA STIFFNESS AND CTE MATRICES HAVE WANT KNOW IF NO T: { } [ Q t ]{ t } t t { } [ Q ]{ } { } [ T ]{ } CL SUB FOR IN TERMS OF X: IF NO {}: t t { } { α } T t t t { } { α } T { α } T t t { } [ T ]{ } t t t t [ T ]{ } [ Q ][ T ]{ } [ T ]{ } { α } T CL CL CL CL MULT THRU BY [T CL ] -, t t [ T ] [ T ]{ } [ T ] [ Q ][ T ]{ } CL CL CL t t { } [ T ] [ Q ][ T ]{ } CL CL COMPARE WITH WANT CL [ ] [ ] [ t t Q T Q ][ T ] CL CL t t [ T ] [ T ]{ } [ T ] { α } T CL CL CL t t { } [ T ] { α } T CL t { α } [ T ] { α t } CL (D-7) ME 75 Lecture 6
62 LAMINA CONSTITUTIVE RELATIONS (cont d) SUMMARY: IN OFF-AXIS (X,Y) LAMINATE COORDINATES HAVE WANT t t { } [ Q ]{ { } { α } T} t t t { } [ Q ]{ { } { α } T} θ TENSOR -D TRANSFORMATION MATRIX TRANSFORMATION EQUATIONS: STRESS AND STRAIN FROM LAMINATE TO LAYER COORDS [ ] T CL m n mn n m { } [ T ]{ } CL mn mn mn m cos θ, n sinθ m n t t { } [ T ]{ } CL STIFFNESS AND CTE MATRICES FROM LAMINATE TO LAYER COORDS [ ] [ ] [ t t Q T Q ][ T ] CL CL t { α } [ T ] { α t } CL ME 75 Lecture 6
63 EXAMPLE T3/58 Gr/Ep LAYER - FIND [] LAYER PLANE STRESS [Q] AND [α] MATRICES ON-AXIS, AND OFF-AXIS FOR 45 AND -45 deg TO X-AXIS X Y Y X X X LAYER MATERIAL PROPERTIES: E E MPa 3 MPa G MPa α.6 α 3 LAYER THICKNESS t.5mm (.8). 59 E C E C ME 75 Lecture 63
64 ME 75 Lecture 64 LAYER MATERIAL AXIS STIFFNESS MATRIX, [Q t ] AND CTE MATRIX, [α] { } C o t α α α [ ] MPa Q t LAYER TRANSFORMATION MATRICES [T CL ] k 45 sin, 45 cos ± ± ± m n [ ] 45 n m mn mn mn m n mn n m T CL [ ] 45 n m mn mn mn m n mn n m T CL [ ] 45 T CL [ ] 45 T CL SIMILARLY, [ ] ( ) ( ) ( ) ( ) G E E E E Q t
65 [ ] LAYER OFF-AXIS STIFFNESS MATRICES Q t k 45 t t [ Q ] [ T ] [ Q ][ T ] 45 CL CL t 3 [ Q ] MPa [ Q ] t MPa SIMILARLY, [ Q ] t MPa ME 75 Lecture 65
66 LAYER OFF-AXIS THERMAL EXPANSION MATRICES { } α k 6 6 (.6 ) ( 3 ) 5.7 / C 6 α m α n α 6 6 ( 3 ) (.6 ) 5.7 / C 6 α n α m α ± 6 6 α α α / mn( ) ( ) C { α } 5.7 /, { α } 5.7 C 45 C / ME 75 Lecture 66
67 IN MATHCAD, LAYER MATERIAL PROPERTIES: E : E : 8 GPa.3 GPa :.8 G : 7.7 GPa newton MPa : mm E : E GPa :.593 MPa α :.6 6 K α : 3 6 K LAYER THICKNESS: t :.5 mm LAYER MATERIAL AXIS STIFFNESS MATRIX [Q t ] CTE MATRIX [α] let V : then Q : E V E V E V E V G Q GPa LAYER CTE MATRIX [α] α : α α α K ME 75 Lecture 67
68 LAYER TRANSFORMATION MATRICES [T CL ] k angle direction cosines ( ) cos( θ deg) m θ : n θ ( ) : sin ( θ deg) m( 45).77 n( 45).77 T( θ) : m θ m( θ) n( θ) ( ) n( θ) m θ n( θ) m( θ) ( ) n ( θ ) ( ) m θ m θ m θ ( ) n( θ) n( θ) ( ) n( θ) T( ) T( 45) T( 45) LAYER MATERIAL OFF-AXIS STIFFNESS MATRICES [Q bar ] Q bar θ ( ) : T( θ) Q T ( θ ) Q bar ( ) GPa Q bar ( 45) GPa Q bar ( 45) GPa ME 75 Lecture 68
69 LAYER CTE MATRICES [α bar ] α bar θ ( ) : T( θ) α α bar ( ) K α bar ( 45) α K bar ( 45) K ME 75 Lecture 69
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