Laminar Composites (eBook)

George Staab (Autor)

eBook Download: PDF | EPUB

2015 | 2. Auflage
466 Seiten
Elsevier Science (Verlag)
978-0-12-802619-9 (ISBN)

Infos

This reference text provides students and practicing engineers with the theoretical knowledge and practical skills needed to identify, model, and solve structural analysis problems involving continuous fiber laminated composites. The principles are illustrated throughout with numerous examples and case studies, as well as example problems similar in nature to those found in strength of materials texts. A solutions manual is available. Extensive coverage of test methods and experimental techniques distinguished Staab from the many theory-led books on composites, making it ideal for practicing engineers and courses with a practical emphasis. The second edition of Laminar Composites is ideal for engineers with a firm understanding of basic structural analysis discovering for the first time the intricacies of orthotropic material behavior and laminate analysis. The fundamental equations required to formulate and assess the behavior of laminated composites are presented in an easy to follow format. Revised and updated throughout, the second edition also includes three new chapters; beams, plates, shells, each covering aspects such as bending, deformation and vibration accompanied by the relevant equations of equilibrium and motion. - Tutorial style ideal for self-study or use on strength of materials courses (undergraduate and graduate - online solutions manual available) - A foundational reference work for a class of composite materials of growing commercial importance - Coverage of test methods and experimental techniques distinguished Staab from the many theory-led books on composites, making it ideal for practicing engineers and courses with a practical emphasis

Industrial experience with Sikorsky Aircraft. Worked with McGraw-Hill on computer-aided undergraduate teaching materials.

Textauszug

2

A review of stress–strain and material behavior

Abstract

This chapter introduces the nomenclature for stresses and strains used in Cartesian, tensor, and material coordinate systems as well as the contracted notation used for laminate analysis. The relationships between strain and displacement are defined and the transformations of strains in one coordinate system to those in another coordinate system are developed. Stresses and stress transformations from one coordinate system to another are also developed. Stress–strain relationships using for a general anisotropic material stiffness matrix are presented. The change in the stiffness matrix for monoclinic, orthotropic, transversely isotropic and isotropic material is presented. A similar presentation for the strain–stress relationships using the compliance matrix is also given. The general effects of temperature and moisture are discussed and complete anisotropic material behavior using compliance and stiffness matrices is presented.

Keywords

Strain

Stress

Anisotropic

Monoclinic

Orthotropic

Transversely Isotropic

Isotropic

Stiffness

Compliance

Thermal expansion

Moisture absorption

2.1 Introduction

In developing methodologies for the analysis and design of laminated composite materials, a consistent nomenclature is required. Stress and strain are presented in terms of Cartesian coordinates for two reason: (1) continuity with developments in undergraduate strength of materials courses and (2) simplification of the analysis procedures involving thermal and hygral effects as well as the general form of the load-strain relationships. A shorthand notation (termed contracted) is used to identify stresses and strains. The coordinate axes are an x–y–z system or a numerical system of 1–2–3. The 1–2–3 system is termed the material or on-axis system. Figure 2.1 shows the relationship between the x–y–z and 1–2–3 coordinate systems. All rotations of coordinate axes are assumed to be about the z-axis, so z is coincident with the three directions, which is consistent with the assumption that individual lamina are modeled as orthotropic materials. The notational relationship between the Cartesian, tensor, material, and contracted stresses and strains is presented below for the special case when the x-, y-, and z-axes coincide with the 1, 2, and 3 axes.

Figure 2.1 Cartesian and material coordinate axes.

x−ɛxσy−ɛyσz−ɛzτyz−γyzτxz−γxzτxy−γxy

xx−ɛxxσyy−ɛyyσzz−ɛzzσyz−2ɛyzσxz−2ɛxzσxy−2ɛxy

1−ɛ1σ2−ɛ2σ3−ɛ3τ23−γ23τ13−γ13τ12−γ12

1−ɛ1σ2−ɛ2σ3−ɛ3σ4−ɛ4σ5−ɛ5σ6−ɛ6

2.2 Strain–displacement relations

When external forces are applied to an elastic body, material points within the body are displaced. If this results in a change in distance between two or more of these points, a deformation exists. A displacement which does not result in distance changes between any two material points is termed a rigid body translation or rotation. The displacement fields for an elastic body can be denoted by U(x, y, z, t), V(x, y, z, t), and W(x, y, z, t), where U, V, and W represent the displacements in the x-, y-, and z-directions, respectively, and t represents time. For the time being, our discussions are limited to static analysis and time is eliminated from the displacement fields. The displacement fields are denoted simply as U, V, and W. For many cases of practical interest, these reduce to planar (two-dimensional) fields.

Assume two adjacent material points A and B in Figure 2.2 are initially a distance dx apart. Assume line AB is parallel to the x-axis and displacements take place in the x–y plane. Upon application of a load, the two points are displaced and occupy new positions denoted as A′ and B′. The change in length of dx is denoted as dL, and is expressed as:

L=dx+∂U∂xdx2+∂V∂xdx2=1+2∂U∂x+∂U∂x2+∂V∂x2dx

Figure 2.2 Displacement of material points A and B.

Assuming U/∂x≪1 and V/∂x≪1, the terms U/∂x2 and V/∂x2 are considered to be zero and the expression for dL becomes:

L=1+2∂U∂xdx

Expanding this in a binomial series, L=1+∂U/∂x+h.o.tdx. The higher order terms (h.o.t.) are neglected since they are small. The normal strain in the x-direction is defined as x=dL−dx/dx. Substituting L=1+∂U/∂x+h.o.tdx, the strain in the x-direction is defined. This approach can be extended to include the y- and z-directions. The resulting relationships are:

x=∂U∂xɛy=∂V∂yɛz=∂W∂z

Shear strain is associated with a net change in right angles of a representative volume element (RVE). The deformation associated with a positive shear is shown in Figure 2.3 for pure shear in the x–y plane. Material points O, A, B, and C deform to O′, A′, B,′ and C′ as shown. Since a condition of pure shear is assumed, the original lengths dx and dy are unchanged. Therefore, U/∂x=0 and V/∂y=0, and the angles θyx and θxy can be defined from the trigonometric relationships

θxy=∂V/∂xdxdxsinθyx=∂U/∂ydydy

Figure 2.3 Deformation under conditions of pure shear.

Small deformations are assumed so approximations of θxy≈θxy and θyx≈θyx are used. The shear strain in the x–y plane is defined as xy=θxy+θyx=π/2−ϕ=∂U/∂y+∂V/∂x, where ϕ represents the angle between two originally orthogonal sides after deformation. For a negative shear strain, the angle ϕ increases. Similar expressions can be established for the x–z and y–z planes. The relationships between shear strain and displacement are:

xy=∂U∂y+∂V∂xγxz=∂U∂z+∂W∂xγyz=∂V∂z+∂W∂y

These are the Cartesian representations of shear strains and are related to the tensor form by: xy=2ɛxy, xz=2ɛxz, and yz=2ɛyz. The 2 in this relationship can make the tensor form of laminate analysis complicated, especially when thermal and hygral effects are considered.

Since strain is directly related to displacement, it is possible to establish the displacement fields U, V, and W from a strain field. For a displacement field to be valid, it must satisfy a set of equations known as the compatibility equations. These equations are generally expressed either in terms of strain or stress components. The compatibility equations ensure that the displacement fields will be single-valued functions of the coordinates when evaluated by integrating displacement gradients along any path in the region. The equations of compatibility can be found in numerous texts on elasticity, such as [1]. The strain component form of the constitutive equations...

Produkt-Details

• Erscheint lt. Verlag: 11.8.2015
• Sprache: englisch
• Themenwelt: Naturwissenschaften ► Chemie ► Physikalische Chemie
• Themenwelt: Technik ► Bauwesen
• Themenwelt: Technik ► Maschinenbau
• ISBN-10: 0-12-802619-7 / 0128026197
• ISBN-13: 978-0-12-802619-9 / 9780128026199