Lamina stiffness plane

In-Plane Shear Properties. The basic lamina in-plane shear stiffness and strength is characterized using a unidirectional hoop-wound (90°) 0.1 -m nominal internal diameter tube that is loaded in torsion. The test method has been standardized under the ASTM D5448 test method for in-plane shear properties of unidirectional fiber-resin composite cylinders. D5448 provides the specimen and hardware geometry necessary to conduct the test. The lamina in-plane shear curve is typically very nonlinear [51]. The test yields the lamina s in-plane shear strength, t12, in-plane shear strain at failure, y12, and in-plane chord shear modulus, G12. 

Coupling terms of laminate stiffness matrix Bending terms of laminate stiffness matrix Longitudinal Young s modulus of the lamina Transverse Young s modulus of the lamina In-plane shear modulus of the lamina Out-of-plane shear modulus of lamina (in the 1-3 plane) Out-of-plane shear modulus of lamina (in 2-3 plane) Moment stress resultants per unit width Force stress resultants per unit width Laminate reduced stiffness terms Transformed reduced stiffness terms... 

The preceding stress-strain and strain-stress relations are the basis for stiffness and stress analysis of an individual lamina subjected to forces in its own plane. Thus, the relations are indispensable in laminate analysis. 

In Section 2.2, the stress-strain relations (generalized Hooke s law) for anisotropic and orthotropic as well as isotropic materials are discussed. These relations have two commonly accepted manners of expression compliances and stiffnesses as coefficients (elastic constants) of the stress-strain relations. The most attractive form of the stress-strain relations for orthotropic materials involves the engineering constants described in Section 2.3. The engineering constants are particularly helpful in describing composite material behavior because they are defined by the use of very obvious and simple physical measurements. Restrictions in the form of bounds are derived for the elastic constants in Section 2.4. These restrictions are useful in understanding the unusual behavior of composite materials relative to conventional isotropic materials. Attention is focused in Section 2.5 on stress-strain relations for an orthotropic material under plane stress conditions, the most common use of a composite lamina. These stress-strain relations are transformed in Section 2.6 to coordinate systems that are not aligned with the principal material... 

The reduced stiffnesses, Qy, are defined in terms of the engineering constants in Equation (2.66). In any other coordinate system in the plane of the lamina, the stresses are... 

The stress-strain relations in arbitrary in-plane coordinates, namely Equation (4.5), are useful in the definition of the laminate stiffnesses because of the arbitrary orientation of the constituent laminae. Both Equations (4.4) and (4.5) can be thought of as stress-strain relations for the k layer of a multilayered laminate. Thus, Equation (4.5) can be written as... 

The invariant stiffness concepts for a iamina will now be extended to a laminate. All results in this and succeeding subsections on invariant laminate stiffnesses were obtained by Tsai and Pagano [7-16 and 7-17]. The laminate is composed of orthotropic laminae with arbitrary orientations and thicknesses. The stiffnesses of the laminate in the x-y plane can be written in the usual manner as... 

All components are given in the principal lamina coordinate system. Inverting eqn 4.2, the in-plane stiffness matrix is obtained ... 

There is an important group of laminates that exhibit in-plane isotropic elastic response. These laminates are called quasi-isotropic. This group includes all symmetric laminates with IN (N > 2) laminae with the same thickness and N equal angles between fibre orientations (A0 = nIN), i.e. = 60° for N = 3, A6 = 45° for N = 4, Ad = 30° for = 6 and so on. It is possible to prove that the in-plane stiffness or extensional matrix of quasi-isotropic laminates is given in reduced form as [14]... 

The Equivalent Constraint Model (ECM) was introduced by Fan and Zhang (1993) with the aim to analyse the in-situ constraint effects on damage evolution in a particular lamina within a multidirectional laminate. In this model, all the laminae below and above the chosen lamina are replaced with homogeneous layers having the equivalent constraining effect. It is assumed that the in-plane stiffness properties of the equivalent constraint layers can be calculated from the classical lamination theory, provided stresses and strains in them are known. 

This stiffness is termed the two-dimensional off-axis reduced stiffness for twisted yarn, because each lamina is loaded in-plane. According to the ideal twist geometry, the twist angle is given as a function of the radius r of the laminated cylinder as shown in Eq. (10.1) or as follows ... 

It is apparent from Equations 8.42 and 8.43 that four material elastic properties (compliance or stiffness) are needed to characterize the in-plane behavior of a linear elastic orthotropic lamina. It is convenient to define these material properties in terms of measured engineering constants (Young s moduli, El > d Ej, shear modulus Glt, and Poisson s ratios u,lt and (Xtl). The longitudinal Young s... 

At this stage, the strains and curvatures are calculated either by Equations 9.41a and 9.41b or by Equation 9.43 that are known inputs. Once the global deformations are known the individnal ply or lamina deformations can be calculated. Remember that since plane sections remain plane, the laminate analysis assumes that strain is both continuous and linearly distributed on a cross section of the laminate. The result of this is that stress is typically discontinuous due to the different reduced stiffness matrices of adjoining plies. Eor example, since the strain in the x-direction at the interface of plies is assumed to be the same, it follows that stress will be different. The total strain as a function of z is... 

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