A review of stress–strain and material behavior

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.

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

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

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...