**Elem Fin**

Apostila de Elementos Finitos, métodos computacionais II

(Parte **1** de 4)

Chapter 1 INTRODUCTION

1.1. Structural materials

Materials are the basic elements of all natural and man-made structures. Figuratively speaking, these materialize the structural conception. Technological progress is associated with continuous improvement of existing material properties as well as with the expansion of structural material classes and types. Usually, new materials emerge due to the necessity to improve structural efficiency and performance. In addition, new materials themselves as a rule, in turn provide new opportunities to develop updated structures and technology, while the latter challenges materials science with new problems and tasks. One of the best manifestations of this interrelated process in the development of materials, structures, and technology is associated with composite materials, to which this book is devoted.

Structural materials possess a great number of physical, chemical and other types of properties, but at least two principal characteristics are of primary importance. These characteristics are the stiffness and strength that provide the structure with the ability to maintain its shape and dimensions under loading or any other external action.

High stiffness means that material exhibits low deformation under loading. However, by saying that stiffness is an important property we do not mean that it should be necessarily high. The ability of a structure to have controlled deformation (compliance) can also be important for some applications (e.g., springs; shock absorbers; pressure, force, and displacement gauges).

Lack of material strength causes an uncontrolled compliance, i.e., in failure after which a structure does not exist any more. Usually, we need to have as high strength as possible, but there are some exceptions (e.g., controlled failure of explosive bolts is used to separate rocket stages).

Thus, without controlled stiffness and strength the structure cannot exist. Naturally, both properties depend greatly on the structure’s design but are determined by the stiffness and strength of the structural material because a good design is only a proper utilization of material properties.

To evaluate material stiffness and strength, consider the simplest test – a bar with crosssectional area A loaded with tensile force F as shown in Fig. 1.1. Obviously, the higher the force causing the bar rupture, the higher is the bar’s strength. However, this strength does not only depend on the material properties – it is proportional to the cross-sectional area A.

2 Advanced mechanics of composite materials A

Fig. 1.1. A bar under tension.

Thus, it is natural to characterize material strength by the ultimate stress

where F is the force causing the bar failure (here and subsequently we use the overbar notation to indicate the ultimate characteristics). As follows from Eq. (1.1), stress is measured as force divided by area, i.e., according to international (SI) units, in pascals (Pa) so that 1Pa =1N/m2. Because the loading of real structures induces relatively high stresses, we also use kilopascals (1kPa = 103 Pa), megapascals (1MPa = 106 Pa), and gigapascals (1GPa = 109 Pa). Conversion of old metric (kilogram per square centimeter) and English (pound per square inch) units to pascals can be done using the following relations: 1kg/cm2 = 98kPa and 1psi = 6.89kPa.

For some special (e.g., aerospace or marine) applications, i.e., for which material density, ρ, is also important, a normalized characteristic

is also used to describe the material. This characteristic is called the ‘specific strength’ of a material. If we use old metric units, i.e., measure force and mass in kilograms and dimensions in meters, substitution of Eq. (1.1) into Eq. (1.2) yields kσ in meters. This result has a simple physical sense, namely kσ is the length of the vertically hanging fiber under which the fiber will be broken by its own weight.

The stiffness of the bar shown in Fig. 1.1 can be characterized by an elongation corresponding to the applied force F or acting stress σ = F/A. However, is proportional to the bar’s length L0. To evaluate material stiffness, we introduce strain

Since ε is very small for structural materials the ratio in Eq. (1.3) is normally multiplied by 100, and ε is expressed as a percentage.

Chapter 1. Introduction 3

Naturally, for any material, there should be some interrelation between stress and strain, i.e.,

These equations specify the so-called constitutive law and are referred to as constitutive equations. They allow us to introduce an important concept of the material model which represents some idealized object possessing only those features of the real material that are essential for the problem under study. The point is that in performing design or analysis we always operate with models rather than with real materials. Particularly, for strength and stiffness analysis, such a model is described by constitutive equations, Eqs. (1.4), and is specified by the form of the function f( σ) or ϕ(ε).

The simplest is the elastic model which implies that f(0) = 0,ϕ (0) = 0 and that

Eqs. (1.4) are the same for the processes of an active loading and an unloading. The corresponding stress–strain diagram (or curve) is presented in Fig. 1.2. The elastic model (or elastic material) is characterized by two important features. First, the corresponding constitutive equations, Eqs. (1.4), do not include time as a parameter. This means that the form of the curve shown in Fig. 1.2 does not depend on the rate of loading (naturally, it should be low enough to neglect inertial and dynamic effects). Second, the active loading and the unloading follow one and the same stress–strain curve as in Fig. 1.2. The work performed by force F in Fig. 1.1 is accumulated in the bar as potential energy, which is also referred to as strain energy or elastic energy. Consider some infinitesimal elongation d and calculate the elementary work performed by the force F in Fig. 1.1 as dW = Fd . Then, work corresponding to point 1 of the curve in Fig. 1.2 is

Fig. 1.2. Stress–strain curve for an elastic material.

4 Advanced mechanics of composite materials where 1 is the elongation of the bar corresponding to point 1 of the curve. The work W is equal to elastic energy of the bar which is proportional to the bar’s volume and can be presented as

0 σdε

is a specific elastic energy (energy accumulated in a unit volume of the bar) that is referred to as an elastic potential. It is important that U does not depend on the history of loading. This means that irrespective of the way we reach point 1 of the curve in Fig. 1.2 (e.g., by means of continuous loading, increasing force F step by step, or using any other loading program), the final value of U will be the same and will depend only on the value of final strain ε1 for the given material. Avery important particular case of the elastic model is the linear elastic model described by the well-known Hooke’s law (see Fig. 1.3)

Here, E is the modulus of elasticity. It follows from Eqs. (1.3) and (1.6), that E = σ if ε = 1, i.e., if = L0. Thus, the modulus can be interpreted as the stress causing elongation of the bar in Fig. 1.1 to be the same as the initial length. Since the majority of structural materials fail before such a high elongation can occur, the modulus is usually much higher than the ultimate stress σ.

Fig. 1.3. Stress–strain diagram for a linear elastic material.

Chapter 1. Introduction 5

Similar to specific strength kσ in Eq. (1.2), we can introduce the corresponding specific modulus

which describes a material’s stiffness with respect to its material density.

Absolute and specific values of mechanical characteristics for typical materials discussed in this book are listed in Table 1.1. After some generalization, the modulus can be used to describe nonlinear material behavior of the type shown in Fig. 1.4. For this purpose, the so-called secant, Es, and tangent, Et, moduli are introduced as

Es = σε = σ

Et = dσ dε

While the slope α in Fig. 1.4 determines the conventional modulus E, the slopes β and γ determine Es and Et, respectively. As can be seen, Es and Et, in contrast to E, depend on the level of loading, i.e., on σ or ε. For a linear elastic material (see Fig. 1.3),

Es = Et = E. Hooke’s law, Eq. (1.6), describes rather well the initial part of stress–strain diagram for the majority of structural materials. However, under a relatively high level of stress or strain, materials exhibit nonlinear behavior.

One of the existing models is the nonlinear elastic material model introduced above (see Fig. 1.2). This model allows us to describe the behavior of highly deformable rubbertype materials.

Another model developed to describe metals is the so-called elastic–plastic material model. The corresponding stress–strain diagram is shown in Fig. 1.5. In contrast to an elastic material (see Fig. 1.2), the processes of active loading and unloading are described with different laws in this case. In addition to elastic strain, εe, which disappears after the load is taken off, the residual strain (for the bar shown in Fig. 1.1, it is plastic strain, εp) remains in the material. As for an elastic material, the stress–strain curve in Fig. 1.5 does not depend on the rate of loading (or time of loading). However, in contrast to an elastic material, the final strain of an elastic–plastic material can depend on the history of loading, i.e., on the law according to which the final value of stress was reached.

Thus, for elastic or elastic–plastic materials, constitutive equations, Eqs. (1.4), do not include time. However, under relatively high temperature practically all the materials demonstrate time-dependent behavior (some of them do it even under room temperature). If we apply some force F to the bar shown in Fig. 1.1 and keep it constant, we can see that for a time-sensitive material the strain increases under a constant force. This phenomenon is called the creep of the material.

So, the most general material model that is used in this book can be described with a constitutive equation of the following type:

6 Advanced mechanics of composite materials

Table 1.1 Mechanical properties of structural materials and fibers.

Material Ultimate tensile stress, σ (MPa)

Modulus,

E (GPa) Specificgravity Maximum specific strength,

Maximum specific modulus,

Metal alloys

Metal wires (diameter, µm)

Thermoset polymeric resins

Thermoplastic polymers

Synthetic fibers

(Parte **1** de 4)