In-plane stiffness matrix

The residual in-plane stiffness matrix [Q] of the equivalent layer in the global co-ordinates can be obtained from the residual in-plane stiffness matrix in the local co-ordinates by... 

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

Here is the in-plane stiffness matrix of layer of the ECM/r laminate, and is the modified in-plane stiffness matrix of the equivalent constraint layer (k layer). Crack surfaces are assumed to be stress-free, so that... 

If there is an infinite numtser of planes of material property symmetry, then the foregoing relations simplify to the isotropic material relations with only two independent constants in the stiffness matrix ... 

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]... 

For symmetric laminates it is possible to define effective in-plane moduli in terms of the in-plane stiffness or extensional compliance matrix, since there is no coupling between in-plane and bending response. The effective... 

The next problem area is transverse shearing effects. There are some distinct characteristics of composite materials that bear very strongly on this situation because for a composite material the transverse shearing stiffness, i.e., perpendicular to the plane of the fibers, is considerably less than the shear stiffness in the plane of the fibers. There is a shear stiffness for a composite material in a plane that involves one fiber direction. Shear involves two directions always, and one of the directions in the plane is a fiber direction. That shear stiffness is quite a bit bigger than the shear stiffness in a plane which is perpendicular to the axis of the fibers. The shear stiffness in a plane which is perpendicular to the axis of the fibers is matrix-dominated and hardly fiber-influenced. Therefore, that shear stiffness is much closer to that of the matrix material itself (a low value compared to the in-plane shear stiffness). 

If the z direction is taken as the normal to the surface, and the x direction is taken as also lying in the plane of incidence, then in calculating the Christoffel matrix T for arbitrary crystallographic orientation the stiffness matrix must be transformed to these new coordinates. In the notation of (11.2), and for an incident wavevector (k x, k y, k z) in the fluid, the requirements of Snell s law become... 

Fracture process in multidirectional composite laminates subjected to in-plane static or fatigue tensile loading involves sequential accumulation of damage in the form of matrix cracks that appear parallel to the fibres in the off-axis plies, edge delamination and local delamination long before catastrophic failure. These resin dominated failure modes significantly reduce the laminate stiffness and are detrimental to its strength. 

In this chapter we use a method that solves Eqn (4.3) using exact, periodic formulations [4]. Here, the eigenvalue analysis is executed on a transcendental stiffness matrix derived from the solution of the governing differential equations of the constituent strips, which are assumed to undergo a deformation that varies sinusoidally to infinity in the longitudinal direction. The out-of-plane buckling displacement w is assumed to be of the form... 

In-plane membrane stiffnesses of a laminate (units force/length) Membrane/bending coupling stiffnesses of a laminate (units force) Measure of crack density in matrix with cracks Inverse of D matrix... 

Due to the extreme difference between the matrix and fiber properties, the in-plane modulus is dominated by the fiber modulus and volume fraction. For example, the GE GEN-IV material has a modulus of 70 GPa. With 15% fiber in the loading direction and a fiber elastic modulus of 380 GPa, the stiffness due to the fibers alone is expected to be 57 GPa. The UCSB Nextel 610-based material with 33% higher fiber content likewise has roughly a 35% higher modulus. The small contribution of the matrix makes it particularly difficult to infer the elastic properties of the matrix from the composite test results [143]. As a result, and because the porous matrix materials are difficult to produce without any fiber reinforcement, their constitutive properties are not yet well characterized [145]. 

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... 

Laminate stiffness analysis predicts the constitutive behaviour of a laminate, based on classical lamination theory (CLT). The result is often given in the form of stiffness and compliance matrices. Engineering constants, i.e. the in-plane and flexural moduli, Poisson s ratios and coefficients of mutual influence, are further derived from the elements of the compliance matrix. Analyses are continuously needed in structural design since it is essential to know the constitutive behaviour of laminates forming the structure. The results are also the necessary input data for all other macromechanical analyses. A computer code for the stiffness analysis is a valuable tool on account of the extensive calculations related to the analysis. 

High matrix and interfacial shear strength are desirable to keep the ineffective fibre length (that part or the fibre not contributing to the strength and stiffness of the composite) as low as possible. Against this, when a composite is loaded in-plane tension only, the interface should be as weak as possible to maximize strength. This situation is nnnsnal, but may occur, for example, in filament-wound pressure vessels. 

ABD stiffness matrix and mechanical deformation for in-plane and flexural loading. 

Finite element models for short fiber reinforced conposites in plane stress and bending have bean developed. Random distribution and orientation of fibers are generated by monte carlo simulation. The entries of the stiffness matrix are treated as means of associated random variables. Two numerical examples demonstrate their behaviour and limitations. Limitations of the present model have also been discussed. 

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