TESTEVACA 1727104811825 Aula_11_-_Principios_de_Macromecnica 1 pdf

UACSA CAMPUS DAS ENGENHARIAS UACSA CAMPUS DAS ENGENHARIAS Aula 11 – Materiais Compósitos UACSA CAMPUS DAS ENGENHARIAS UACSA CAMPUS DAS ENGENHARIAS Princípios Básicos de Macromecânica Aplicados a Compósitos Estruturais Aula 11 - Materiais Compósitos 2 UACSA CAMPUS DAS ENGENHARIAS UACSA CAMPUS DAS ENGENHARIAS Terminologia – Mecânica dos compósitos 3 Fibra Matriz Micromecânica de uma Lâmina Macromecânica de uma Lâmina Macromecânica de um Laminado UACSA CAMPUS DAS ENGENHARIAS UACSA CAMPUS DAS ENGENHARIAS Deformation of Isotropic Lamina = δ δ = δ δ B A B A 1 2 2 1 , FIGURE 2.2 Deformation of square, isotropic plate under normal loads UACSA CAMPUS DAS ENGENHARIAS UACSA CAMPUS DAS ENGENHARIAS Deformation of Unidirectional Lamina δ δ δ δ B A B A 1 2 2 1 ,   FIGURE 2.2 Deformation of square, unidirectional lamina with fibers at zero angle under normal loads UACSA CAMPUS DAS ENGENHARIAS UACSA CAMPUS DAS ENGENHARIAS Deformation of Unidirectional Angled Lamina FIGURE 2.4 Deformation of square, unidirectional lamina with fibers at an angle to normal loads UACSA CAMPUS DAS ENGENHARIAS UACSA CAMPUS DAS ENGENHARIAS Composite Analysis 2D vs. 3D State of Stress 7 UACSA CAMPUS DAS ENGENHARIAS TTeennssoorr tteennssããoo e vveettoorr tteennssããoo UACSA CAMPUS DAS ENGENHARIAS UACSA CAMPUS DAS ENGENHARIAS Estado de tensão num ponto •Tensão é uma grandeza tensorial: [σ] - chamado tensor de tensões; •Uma vez conhecidas as nove componentes do tensor de tensões, pode-se determinar o vetor tensão atuando sobre qualquer plano que passa pelo ponto; UACSA CAMPUS DAS ENGENHARIAS UACSA CAMPUS DAS ENGENHARIAS Estado uniaxial de tensão (1D) Ex. – ensaio de tração =  0 0  0 0  x 0 0  0  0 UACSA CAMPUS DAS ENGENHARIAS UACSA CAMPUS DAS ENGENHARIAS Estado plano de tensões (2D) Ex. – peças de pequena espessura σ= yx  0 σx  xy 0  σy 0 0 0 UACSA CAMPUS DAS ENGENHARIAS UACSA CAMPUS DAS ENGENHARIAS Estado triplo de tensões (3D) σ x σ=  yx  zx   xy xz σ y yz  zy σ z     UACSA CAMPUS DAS ENGENHARIAS UACSA CAMPUS DAS ENGENHARIAS Transformação de tensão para o estado plano UACSA CAMPUS DAS ENGENHARIAS UACSA CAMPUS DAS ENGENHARIAS Tensões principais 1 y x xy yx 1 2 UACSA CAMPUS DAS ENGENHARIAS 15 PROPRIEDADES ELÁSTICAS – COEFICIENTE DE POISSON Coeficiente Poison UACSA CAMPUS DAS ENGENHARIAS UACSA CAMPUS DAS ENGENHARIAS Lâminas 16 Aula 11 – Materiais Compósitos UACSA CAMPUS DAS ENGENHARIAS UACSA CAMPUS DAS ENGENHARIAS Macromechanics Study of stress-strain behavior of composites using effective properties of an equivalent homogeneous material. Only the globally averaged stresses and strains are considered, not the local fiber and matrix values. Estudo do comportamento tensão-deformação de compósitos utilizando propriedades efetivas de um material homogêneo equivalente. Apenas as tensões e tensões médias globais são consideradas, não os valores locais nas fibra e matriz UACSA CAMPUS DAS ENGENHARIAS UACSA CAMPUS DAS ENGENHARIAS Stress-Strain Relationships for Anisotropic Materials First, we discuss the form of the stress-strain relationships at a point within the material, then discuss the concept of effective moduli for heterogeneous materials where properties may vary from point-to-point. Primeiro, discutimos a forma das relações tensão- deformação num ponto dentro do material, depois discutimos o conceito de módulos efetivos para materiais heterogêneos onde as propriedades podem variar de ponto a ponto. UACSA CAMPUS DAS ENGENHARIAS UACSA CAMPUS DAS ENGENHARIAS General Form of Elastic - Relationships for Constant Environmental Conditions Each component of stress, ij, is related to each of nine strain components, ij (Note: These relationships may be nonlinear) 11 12 13 ( , , ,...), , 1,2,3... (2.1) ij Fij i j     = = UACSA CAMPUS DAS ENGENHARIAS UACSA CAMPUS DAS ENGENHARIAS 3D state of stress UACSA CAMPUS DAS ENGENHARIAS UACSA CAMPUS DAS ENGENHARIAS Symmetry Simplifies the Generalized Hooke’s Law 1. Symmetry of shear stresses and strains: Same condition for shear strains, 2. Material property symmetry – several types will be discussed. O 1 2 21 12 0 0   = =  implies M Equlibrium Static ji ij ji ij or in general or     = = , ji ij   = 21  12 UACSA CAMPUS DAS ENGENHARIAS UACSA CAMPUS DAS ENGENHARIAS Shear Stress – Tensão cisalhamento = τ , τ yx xy = τ , τ zy yz = τ τ xz zx FIGURE 2.7 Stresses on an infinitesimal cuboid UACSA CAMPUS DAS ENGENHARIAS UACSA CAMPUS DAS ENGENHARIAS Strain- Deformação FIGURE 2.8 Normal and shearing strains on an infinitesimal area in the x-y plane UACSA CAMPUS DAS ENGENHARIAS Geometry of Shear Strain 2 xy xy   = 2 xy xy   =  xy = Engineering Strain  xy = Tensor Strain Total change in original angle =  xy Amount each edge rotates =  xy/2 =  xy UACSA CAMPUS DAS ENGENHARIAS ji ij ji ij and     = = Symmetry of shear stresses and shear strains: Thus, only 6 components of ij are independent, and likewise for ij. This leads to a contracted notation. UACSA CAMPUS DAS ENGENHARIAS Stresses Tensor Notation Contracted Notation 11 1 22 2 33 3 23= 32 4 13= 31 5 12= 21 6 12 ou 13 ou 23 ou 4 ou 5 ou 6 ou UACSA CAMPUS DAS ENGENHARIAS Strains Tensor Notation Contracted Notation  11  1  22  2  33  3 2 23= 2 32=  23=  32  4 2 13= 2 31=  13=  31  5 2 12= 2 21=  12= 21  6 UACSA CAMPUS DAS ENGENHARIAS UACSA CAMPUS DAS ENGENHARIAS Using contracted notation or in matrix form where and are column vectors and [C] is a 6x6 matrix (the stiffness matrix) 2,1 ,...,6 , = = i j C j ij i          C =       Expanding: Stiffness Matrix [C] UACSA CAMPUS DAS ENGENHARIAS Form for anisotropic, with 36 coefficients, Expanding: Stiffness Matrix [C]                                                                     γ γ γ ε ε ε 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 = τ τ τ σ σ σ 12 31 23 3 2 1 66 65 64 63 62 61 56 55 54 53 52 51 46 45 44 43 42 41 36 35 34 33 32 31 26 25 24 23 22 21 16 15 14 13 12 11 12 31 23 3 2 1 UACSA CAMPUS DAS ENGENHARIAS 2,1 ,...,6 , = = i j S j ij i          S =     = C −1 S (2.5) (2.6) Alternatively, or where [S] = compliance matrix and Expanding: Compliance Matrix [S] UACSA CAMPUS DAS ENGENHARIAS Form for anisotropic, with 36 coefficients, Expanding: Compliance Matrix [S]                                                                     τ τ τ σ σ σ 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 = γ γ γ ε ε ε 12 31 23 3 2 1 66 65 64 63 62 61 56 55 54 53 52 51 46 45 44 43 42 41 36 35 34 33 32 31 26 25 24 23 22 21 16 15 14 13 12 11 12 31 23 3 2 1 UACSA CAMPUS DAS ENGENHARIAS UACSA CAMPUS DAS ENGENHARIAS • Up to now, we only considered the stresses and strains at a point within the material, and the corresponding elastic constants at a point. • What do we do in the case of a composite material, where the properties may vary from point to point? • Use the concept of effective moduli of an equivalent homogeneous material. UACSA CAMPUS DAS ENGENHARIAS 2  2  d 2 2  2  2  2  2  2 2  Stress Strain L Heterogeneous composite under varying stresses and strains Equivalent homogeneous material under average stresses and strains Concept of an Effective Modulus of an Equivalent Homogeneous Material Stress, Strain, 3x 3x 3x 3x UACSA CAMPUS DAS ENGENHARIAS 34 3-D Case, Orthotropic and Isotropic UACSA CAMPUS DAS ENGENHARIAS 3-D Case, Specially Orthotropic 1 2 3 1, 2 , 3 principal material coordinates UACSA CAMPUS DAS ENGENHARIAS 3-D Case, Specially Orthotropic UACSA CAMPUS DAS ENGENHARIAS 1  1  2  2  12  12  (a) (b) (c) Simple states of stress used to define lamina engineering constants for specially orthotropic lamina. 3-D Case, Specially Orthotropic UACSA CAMPUS DAS ENGENHARIAS Consider normal stress 1 alone: 1 1 1 2 3 Resulting strains, ; 1 1 1 E   = 1 1 12 12 1 2 E      = − = − (2.19) 3-D Case, Specially Orthotropic UACSA CAMPUS DAS ENGENHARIAS Typical stress-strain curves from ASTM D3039 tensile tests Stress-strain data from longitudinal tensile test of carbon/epoxy composite. UACSA CAMPUS DAS ENGENHARIAS 1 1 13 13 1 3 E      = − − = where E1 = longitudinal modulus ij = Poisson’s ratio for strain along j direction due to loading along i direction Similarly, 3-D Case, Specially Orthotropic UACSA CAMPUS DAS ENGENHARIAS 2  2  1 2 3 Now consider normal stress 2 alone: Strains: ; 2 2 2 E   = 2 1 21 2 21 E2   = −  = − (2.20) 2 3 23 2 23 E2   = −  = − Where E2 = transverse modulus 3-D Case, Specially Orthotropic UACSA CAMPUS DAS ENGENHARIAS UACSA CAMPUS DAS ENGENHARIAS • Observation: All shear strains are zero under pure normal stress (no shear coupling). 0 23 13 12 = = =     For 3 2 1 , ,    alone 3-D Case, Specially Orthotropic UACSA CAMPUS DAS ENGENHARIAS Now, consider shear stress alone, 1 2 3 12  12 12   = Strain 12 12 12 G   = Where G12 = Shear modulus in 1-2 plane 0 23 13 3 2 1 = = = = =      (No shear coupling) (2.21) 3-D Case, Specially Orthotropic UACSA CAMPUS DAS ENGENHARIAS Similarly, for alone 13  ; 13 13 13 G   = 0 23 12 3 2 1 = = = = =      and for alone 23  ; 23 23 23 G   = 0 12 13 3 2 1 = = = = =      Now add strains due to all stresses using superposition 3-D Case, Specially Orthotropic UACSA CAMPUS DAS ENGENHARIAS 31 21 1 2 3 32 12 1 1 1 2 3 2 2 13 23 3 3 1 2 3 23 23 23 31 31 12 12 31 12 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 E E E E E E E E E G G G     − −        −  −                         − −               =                                                   12 coefficients, but only are 9 independent 3-D Orthotropic – strains in terms of stresses Compliance Matrix [S] UACSA CAMPUS DAS ENGENHARIAS Symmetry: ji ij S S = ij ji i j E E    = Only 9 independent coefficients.  Generally orthotropic 3-D case – similar to anisotropic with 36 nonzero coefficients, but 9 are independent as with specially orthotropic case UACSA CAMPUS DAS ENGENHARIAS Specially Orthotropic – Transversely Isotropic 1 2 3 Fibers randomly packed in 2-3 plane, so properties are invariant to rotation about 1- axis (2 same as 3) UACSA CAMPUS DAS ENGENHARIAS Specially orthotropic, transversely isotropic (axis 2 and 3 interchangeable) 12 , 13 G G = 3, 2 E = E 21 31  =  2 23 32 2(1 ) E G = +  Now, only 5 coefficients are independent. Specially Orthotropic – Transversely Isotropic UACSA CAMPUS DAS ENGENHARIAS 3-D Case, Isotropic 1 2 3 1, 2 , 3 principal material coordinates No fiber preferred direction UACSA CAMPUS DAS ENGENHARIAS Isotropic 13 23 12 1 2 3 12 23 13 2(1 ) G G G G E E E E E G = = = = = =  =  =  =  = +  2 independent coefficients Usually measure E, υ – calculate G UACSA CAMPUS DAS ENGENHARIAS                                                                   − − − − − −                         τ τ τ σ σ σ G G G E E E E E E E E E = γ γ γ ε ε ε 12 31 23 3 2 1 12 31 23 3 2 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1       3-D Isotropic – strains in terms of stresses Compliance Matrix [S] UACSA CAMPUS DAS ENGENHARIAS Elastic coefficients in the stress-strain relationship for different materials and coordinate systems Material and coordinate system Number of nonzero coefficients Number of independent coefficients Three – dimensional case Anisotropic 36 21 Generally Orthotropic (nonprincipal coordinates) 36 9 Specially Orthotropic (Principal coordinates) 12 9 Specially Orthotropic, transversely isotropic 12 5 Isotropic 12 2 Two – dimensional case (lamina) Anisotropic 9 6 Generally Orthotropic (nonprincipal coordinates) 9 4 Specially Orthotropic (Principal coordinates) 5 4 Balanced orthotropic, or square symmetric (principal coordinates) 5 3 Isotropic 5 2 UACSA CAMPUS DAS ENGENHARIAS 53 2-D Case, Orthotropic A unidirectional lamina falls under the orthotropic material category. If the lamina is thin and does not carry any out-of- plane loads, one can assume plane stress conditions for the lamina. UACSA CAMPUS DAS ENGENHARIAS 2-D Cases - Simplification Use 3-D equations with, 0 23 13 3 = = =    Plane stress, 0 , , , 12 2 1     0 , , ,  xy y x    Or UACSA CAMPUS DAS ENGENHARIAS UACSA CAMPUS DAS ENGENHARIAS Pure Axial Load in Direction 1 Apply a pure axial load in direction 1 0 0 0 12 2 1 , σ = , τ = σ                                      τ σ σ S S S S S = γ ε ε 12 2 1 66 22 12 12 11 12 2 1 0 0 0 0 S = ε σ E 11 1 1 1 1  . S S = ε ε ν 11 12 1 2 12 − −  E = S 1 11 1 E ν = S 1 12 12 − 0 12 1 12 2 1 11 1 = γ =S σ ε ε =S σ UACSA CAMPUS DAS ENGENHARIAS UACSA CAMPUS DAS ENGENHARIAS Pure Axial Load in Direction 2 Apply a pure axial load in direction 2 0 0 0 12 2 1 , τ = , σ σ  =                                     τ σ σ S S S S S = γ ε ε 12 2 1 66 22 12 12 11 12 2 1 0 0 0 0 S = ε σ E 22 2 2 2 1  . S S = ε ε ν 22 12 2 1 21 − −  E = S 2 22 1 E ν = S 2 21 12 − 0 12 2 22 2 2 12 1 = γ σ =S ε ε =S σ UACSA CAMPUS DAS ENGENHARIAS Reciprocal Relationship E ν = S 2 21 12 − E ν = S 1 12 12 − E ν = E ν 2 21 1 12 UACSA CAMPUS DAS ENGENHARIAS UACSA CAMPUS DAS ENGENHARIAS Pure Shear Load in Plane 12 Apply a pure shear load in Plane 12 0 0 0 12 2 1  = , σ = , τ σ                                     τ σ σ S S S S S = γ ε ε 12 2 1 66 22 12 12 11 12 2 1 0 0 0 0 12 66 12 2 1 0 0 =S  γ = ε = ε S = G 66 12 12 12 1    12 66 1 S = G UACSA CAMPUS DAS ENGENHARIAS UACSA CAMPUS DAS ENGENHARIAS Compliance Matrix – 2 D case                                     τ σ σ S S S S S = γ ε ε 12 2 1 66 22 12 12 11 12 2 1 0 0 0 0                               − −             τ σ σ G E E ν E ν E = γ ε ε 12 2 1 12 2 1 12 1 12 1 12 2 1 1 0 0 0 1 0 1 2 21 UACSA CAMPUS DAS ENGENHARIAS UACSA CAMPUS DAS ENGENHARIAS Coefficients of Stiffness Matrix                                     γ ε ε Q Q Q Q Q = τ σ σ 12 2 1 66 22 12 12 11 12 2 1 0 0 0 0 S = Q S S S S = Q S S S S = Q S S S S = Q 66 66 2 12 22 11 11 22 2 12 22 11 12 12 2 12 22 11 22 11 1 − − − −                                   − − −             γ ε ε G ν ν E ν ν E ν ν ν E ν ν ν - E = τ σ σ 12 2 1 12 12 21 2 12 21 2 12 12 21 2 12 12 21 1 12 2 1 0 0 0 1 1 0 1 1 TABLE 2.2 Typical Values of Lamina Engineering Constants for Several Composites Having Fiber Volume Fraction vf Material E1 (Msi [GPa]) E2 (Msi [GPa]) G12 (Msi [GPa]) v12 vf T300/934 carbon/epoxy 19.0 (131) 1.5 (10.3) 1.0 (6.9) 0.22 0.65 AS/3501 carbon/epoxy 20.0 (138) 1.3 (9.0) 1.0 (6.9) 0.3 0.65 P-100/ERL 1962 pitch/ carbon/epoxy 68.0 (468.9) 0.9 (6.2) 0.81 (5.58) 0.31 0.62 IM7/8551-7 carbon/ toughened epoxy 23.5(162) 1.21(8.34) 0.3(2.07) 0.34 0.6 AS4/APC2 carbon/ PEEK 19.1(131) 1.26(8.7) 0.73(5.0) 0.28 0.58 Boron/6061 boron/ aluminum 34.1(235) 19.9(137) 6.8(47.0) 0.3 0.5 Kevlar® 49/934 aramid/ epoxy 11.0 (75.8) 0.8 (5.5) 0.33 (2.3) 0.34 0.65 Scotchply® 1002 E-glass/epoxy 5.6 (38.6) 1.2 (8.27) 0.6 (4.14) 0.26 0.45 Boron/5505 boron/ epoxy 29.6 (204.0) 2.68 (18.5) 0.81 (5.59) 0.23 0.5 Spectra® 900/826 polyethylene/epoxy 4.45 (30.7) 0.51 (3.52) 0.21 (1.45) 0.32 0.65 E-glass/470-36 E-glass/vinylester 3.54 (24.4) 1.0 (6.87) 0.42 (2.89) 0.32 0.30 Kevlar® is a registered trademark of DuPont Company, Wilmington, Delaware; Scotch- ply® is a registered trademark of 3M Company, St. Paul, Minnesota; and Spectra® is a registered trademark of Honeywell International, Inc. UACSA CAMPUS DAS ENGENHARIAS UACSA CAMPUS DAS ENGENHARIAS Hooke's Law for a 2D Angle Lamina FIGURE 2.20 Local and global axes of an angle lamina UACSA CAMPUS DAS ENGENHARIAS Sign convention for lamina orientation  + x y 2 1  − x y 2 1 Positive  Negative  UACSA CAMPUS DAS ENGENHARIAS Transformation of Stress and Strain UACSA CAMPUS DAS ENGENHARIAS Area Forces Stresses UACSA CAMPUS DAS ENGENHARIAS These three equilibrium equations may be combined in matrix form as follows: UACSA CAMPUS DAS ENGENHARIAS UACSA CAMPUS DAS ENGENHARIAS Relationship of Global and Local Stresses                                                             − −             τ σ σ c - s -sc sc sc c s - sc s c = τ σ σ τ σ σ c - s sc sc sc c s sc s c = τ σ σ xy y x xy y x 12 2 1 2 2 2 2 2 2 2 2 2 2 2 2 12 2 1 2 2 2 2                                                 − τ σ σ T = τ σ σ τ σ σ T = τ σ σ xy y x xy y x 12 2 1 1 12 2 1 ] [ [ ] UACSA CAMPUS DAS ENGENHARIAS UACSA CAMPUS DAS ENGENHARIAS Invariant Form of Stiffness Matrix , Q Q Q Q Q Q Q Q Q = xy y x 66 26 16 26 22 12 16 12 11 xy y x                                             θ θ+U Q = U +U cos4 cos2 3 2 1 11 θ U Q =U 3 cos4 4 12 − θ θ+U U Q =U cos4 cos2 3 2 1 22 − θ θ+U Q =U sin 4 sin 2 2 3 2 16 θ U θ Q =U sin 4 sin 2 2 3 2 26 − θ U U U Q = cos4 ) 2 ( 1 3 4 1 66 − − ) 4 2 3 8 (3 1 66 12 22 11 1 Q + Q + Q + Q = U ) 2 ( 1 22 11 2 Q Q U = − ) 4 2 8 ( 1 66 12 22 11 3 Q Q Q +Q U = − − ) 4 6 8 ( 1 66 12 22 11 4 Q Q +Q + Q U = − UACSA CAMPUS DAS ENGENHARIAS UACSA CAMPUS DAS ENGENHARIAS Invariant Form of Compliance Matrix , S S S S S S S S S = xy y x 66 26 16 26 22 12 16 12 11 xy y x                                           θ, θ + V S = V +V cos4 cos2 3 2 1 11 4 , V =V S 3 4 12 Cos  − 4 , 2 +V V =V S 3 2 1 22   Cos Cos − θ, θ + V S = V sin 4 2 sin 2 3 2 16 4 , 2 2 V =V S 3 2 26   Sin Sin − 4 , 4 V ) = 2(V V S 3 4 1 66 Cos  − − ), 8 (3S +3 S + 2 S + S V = 1 66 12 22 11 1 ), 2 (S S V = 1 22 11 2 − ), 2 S S 8 ( S + S V = 1 66 12 22 11 3 − − ) S +6 S 8 ( S + S V = 1 66 12 22 11 4 − UACSA CAMPUS DAS ENGENHARIAS UACSA CAMPUS DAS ENGENHARIAS Efeito da Orientação Resistência a tração 1(L) y x 2 (T) +  UACSA CAMPUS DAS ENGENHARIAS UACSA CAMPUS DAS ENGENHARIAS 71 Efeito da Orientação Modulo Elástico e Cisalhamento UACSA CAMPUS DAS ENGENHARIAS UACSA CAMPUS DAS ENGENHARIAS Laminados 72 Aula 11 – Materiais Compósitos UACSA CAMPUS DAS ENGENHARIAS UACSA CAMPUS DAS ENGENHARIAS Design and analysis of composite laminates: Laminated Plate Theory (LPT) • Used to determine the response of a composite laminate based on properties of a layer (or ply) UACSA CAMPUS DAS ENGENHARIAS UACSA CAMPUS DAS ENGENHARIAS Laminate Ply Orientation Code • Designate each ply by it’s fiber orientation angle • List plies in sequence starting from top of laminate • Adjacent plies are separated by “/” if their angle is different • Designate groups of plies with same angle using subscripts • Enclose complete laminate in brackets • Use subscript “S” to denote mid plane symmetry, or “T” to denote total laminate • Bar on the top of the ply indicates mid-plane UACSA CAMPUS DAS ENGENHARIAS UACSA CAMPUS DAS ENGENHARIAS UACSA CAMPUS DAS ENGENHARIAS UACSA CAMPUS DAS ENGENHARIAS Special types of laminates • Symmetric laminate – for every ply above the laminate mid plane, there is an identical ply(material and orientation) an equal distance below the mid plane • Balanced laminate – for every ply at a +θ orientation, there is another ply at the – θ orientation somewhere in the laminate • Cross-ply laminate – composed of plies of either 0o or 90o (no other ply orientations) • Quasi-isotropic laminate – produced using at least three different ply orientations, all with equal angles between them. Exhibits isotropic extensional stiffness properties (have the same E in all in-plane directions) UACSA CAMPUS DAS ENGENHARIAS UACSA CAMPUS DAS ENGENHARIAS The response of special laminates • Balanced, unsymmetric, laminate • Tensile loading produces twisting curvature • Ex: [+θ/0/- θ]τ • Symmetric, unbalanced laminate • Tensile loading produces in-plane shearing • Ex: [+θ/0/- θ]τ • Unsymmetric cross-ply laminate • Tensile loading produces bending curvature • Ex: [0/90]τ • Balanced and symmetric laminate • Tensile loading produces extension • Ex: [+θ/- θ]s • Quasi isotropic laminate: [+60/0/- 60]s and [+45/0/+45/90]s • Tensile loading produces extension loading, independent of angle