# Introduction to Mechanics for Solids and Structures (Finite Element Method) Part 1

Introduction

This topic tries to introduce these basic concepts and classical theories in a brief and easy to understand manner. Solids and structures are stressed when they are subjected to loads or forces. The stresses are, in general, not uniform, and lead to strains, which can be observed as either deformation or displacement. Solid mechanics and structural mechanics deal with the relationships between stresses and strains, displacements and forces, stresses (strains) and forces for given boundary conditions of solids and structures. These relationships are vitally important in modelling, simulating and designing engineered structural systems.

Forces can be static and/or dynamic. Statics deals with the mechanics of solids and structures subjected to static loads such as the deadweight on the floor of buildings. Solids and structures will experience vibration under the action of dynamic forces varying with time, such as excitation forces generated by a running machine on the floor. In this case, the stress, strain and displacement will be functions of time, and the principles and theories of dynamics must apply. As statics can be treated as a special case of dynamics, the static equations can be derived by simply dropping out the dynamic terms in the general, dynamic equations. This topic will adopt this approach of deriving the dynamic equation first, and obtaining the static equations directly from the dynamic equations derived.

Depending on the property of the material, solids can be elastic, meaning that the deformation in the solids disappears fully if it is unloaded. There are also solids that are considered plastic, meaning that the deformation in the solids cannot be fully recovered when it is unloaded. Elasticity deals with solids and structures of elastic materials, and plasticity deals with those of plastic materials. The scope of this topic deals mainly with solids and structures of elastic materials. In addition, this topic deals only with problems of very small deformation, where the deformation and load has a linear relationship. Therefore, our problems will mostly be linear elastic.

Materials can be anisotropic, meaning that the material property varies with direction. Deformation in anisotropic material caused by a force applied in a particular direction may be different from that caused by the same force applied in another direction. Composite materials are often anisotropic. Many material constants have to be used to define the material property of anisotropic materials. Many engineering materials are, however, isotropic, where the material property is not direction-dependent. Isotropic materials are a special case of anisotropic material. There are only two independent material constants for isotropic material. Usually, the two most commonly used material constants are the Young’s modulus and the Poisson’s ratio. This topic deals mostly with isotropic materials. Nevertheless, most of the formulations are also applicable to anisotropic materials.

Boundary conditions are another important consideration in mechanics. There are displacement and force boundary conditions for solids and structures. For heat transfer problems there are temperature and convection boundary conditions.

Structures are made of structural components that are in turn made of solids. There are generally four most commonly used structural components: truss, beam, plate, and shell, as shown in Figure 2.1. In physical structures, the main purpose of using these structural components is to effectively utilize the material and reduce the weight and cost of the structure. A practical structure can consist of different types of structural components, including solid blocks. Theoretically, the principles and methodology in solid mechanics can be applied to solve a mechanics problem for all structural components, but this is usually not a very efficient method. Theories and formulations for taking geometrical advantages of the structural components have therefore been developed. Formulations for a truss, a beam, 2D solids and plate structures will be discussed in this topic. In engineering practice, plate elements are often used together with two-dimensional solids for modelling shells. Therefore in this topic, shell structures will be modelled by combining plate elements and 2D solid elements.

## Equations For Three-Dimensional Solids

### Stress and Strain

Let us consider a continuous three-dimensional (3D) elastic solid with a volume V and a surface S, as shown in Figure 2.2. The surface of the solid is further divided into two types of surfaces: a surface on which the external forces are prescribed is denoted Sf ; and surface on which the displacements are prescribed is denoted Sd. The solid can also be loaded by body force f b and surface force fs in any distributed fashion in the volume of the solid.

At any point in the solid, the components of stress are indicated on the surface of an ‘infinitely’ small cubic volume, as shown in Figure 2.3. On each surface, there will be the normal stress component, and two components of shearing stress. The sign convention for the subscript is that the first letter represents the surface on which the stress is acting, and the second letter represents the direction of the stress. The directions of the stresses shown in the figure are taken to be the positive directions. By taking moments of forces about the central axes of the cube at the state of equilibrium, it is easy to confirm that

Figure 2.1. Four common types of structural components. Their geometrical features are made use of to derive dimension reduced system equations.

Therefore, there are six stress components in total at a point in solids. These stresses are often called a stress tensor. They are often written in a vector form of

Corresponding to the six stress tensors, there are six strain components at any point in a solid, which can also be written in a similar vector form of

Figure 2.2. Solid subjected to forces applied within the solid (body force) and on the surface of the solid (surface force).

Figure 2.3. Six independent stress components at a point in a solid viewed on the surfaces of an infinitely small cubic block.

Strain is the change of displacement per unit length, and therefore the components of strain can be obtained from the derivatives of the displacements as follows:

where u, v and w are the displacement components in the x,y and z directions, respectively. The six strain-displacement relationships in Eq. (2.4) can be rewritten in the following matrix form:

where U is the displacement vector, and has the form of

and L is a matrix of partial differential operators obtained simply by inspection on Eq. (2.4):

### Constitutive Equations

The constitutive equation gives the relationship between the stress and strain in the material of a solid. It is often termed Hooke’s law. The generalised Hooke’s law for general anisotropic materials can be given in the following matrix form:

where c is a matrix of material constants, which are normally obtained through experiments. The constitutive equation can be written explicitly as

Note that, sincethere    are    altogether    21    independent material constants which is the case for a fully anisotropic material. For isotropic materials, however, c can be reduced to

where

in which E, ν and G are Young’s modulus, Poisson’s ratio, and the shear modulus of the material, respectively. There are only two independent constants among these three constants. The relationship between these three constants is

That is to say, for any isotropic material, given any two of the three constants, the other one can be calculated using the above equation.

### Dynamic Equilibrium Equation

To formulate the dynamic equilibrium equations, let us consider an infinitely small block of solid, as shown in Figure 2.4. As in forming all equilibrium equations, equilibrium of forces is required in all directions. Note that, since this is a general, dynamic system, we have to consider the inertial forces of the block. The equilibrium of forces in the x direction gives

Figure 2.4. Stresses on an infinitely small block. Equilibrium equations are derived based on this state of stresses.

where the term on the right-hand side of the equation is the inertial force term, and fx is the external body force applied at the centre of the small block. Note that

Hence, Eq. (2.13) becomes one of the equilibrium equations, written as

Similarly, the equilibrium of forces in the y and z directions results in two other equilibrium equations:

The equilibrium equations, Eqs. (2.15) to (2.17), can be written in a concise matrix form

where fb is the vector of external body forces in the x, y and z directions:

Using Eqs. (2.5) and (2.8), the equilibrium equation Eq. (2.18) can be further written in terms of displacements:

The above is the general form of the dynamic equilibrium equation expressed as a matrix equation. If the loads applied on the solid are static, the only concern is then the static status of the solid. Hence, the static equilibrium equation can be obtained simply by dropping the dynamic term in Eq. (2.20), which is the inertial force term:

### Boundary Conditions

There are two types of boundary conditions: displacement (essential) and force (natural) boundary conditions. The displacement boundary condition can be simply written as

on displacement boundaries. The bar stands for the prescribed value for the displacement component. For most of the actual simulations, the displacement is used to describe the support or constraints on the solid, and hence the prescribed displacement values are often zero. In such cases, the boundary condition is termed as a homogenous boundary condition. Otherwise, they are inhomogeneous boundary conditions.

The force boundary condition are often written as

on force boundaries, where n is given by

in which ni (i = x, y, z) are cosines of the outwards normal on the boundary. The bar stands for the prescribed value for the force component. A force boundary condition can also be both homogenous and inhomogeneous. If the condition is homogeneous, it implies that the boundary is a free surface.

The reader may naturally ask why the displacement boundary condition is called an essential boundary condition and the force boundary condition is called a natural boundary conditions. The terms ‘essential’ and ‘natural’ come from the use of the so-called weak form formulation (such as the weighted residual method) for deriving system equations. In such a formulation process, the displacement condition has to be satisfied first before derivation starts, or the process will fail. Therefore, the displacement condition is essential. As long as the essential (displacement) condition is satisfied, the process will lead to the equilibrium equations as well as the force boundary conditions. This means that the force boundary condition is naturally derived from the process, and it is therefore called the natural boundary condition. Since the terms essential and natural boundary do not describe the physical meaning of the problem, it is actually a mathematical term, and they are also used for problems other than in mechanics.

Equations obtained in this section are applicable to 3D solids. The objective of most analysts is to solve the equilibrium equations and obtain the solution of the field variable, which in this case is the displacement. Theoretically, these equations can be applied to all other types of structures such as trusses, beams, plates and shells, because physically they are all 3D in nature. However, treating all the structural components as 3D solids makes computation very expensive, and sometimes practically impossible. Therefore, theories for taking geometrical advantage of different types of solids and structural components have been developed. Application of these theories in a proper manner can reduce the analytical and computational effort drastically. A brief description of these theories is given in the following sections.

## Equations for Two-Dimensional Solids

### Stress and Strain

Three-dimensional problems can be drastically simplified if they can be treated as a two dimensional (2D) solid. For representation as a 2D solid, we basically try to remove one coordinate (usually the z-axis), and hence assume that all the dependent variables are independent of the z-axis, and all the external loads are independent of the z coordinate, and applied only in the x-y plane. Therefore, we are left with a system with only two coordinates, the x and the y coordinates. There are primarily two types of 2D solids. One is a plane stress solid, and another is a plane strain solid. Plane stress solids are solids whose thickness in the z direction is very small compared with dimensions in the x and y directions. External forces are applied only in the x-y plane, and stresses in the z direction are all zero, as shown in Figure 2.5. Plane strain solids are those solids whose thickness in the z direction is very large compared with the dimensions in the x and y directions. External forces are applied evenly along the z axis, and the movement in the z direction at any point is constrained. The strain components in the z direction are, therefore, all zero, as shown in Figure 2.6.

Note that for the plane stress problems, the strains εχι and εyz are zero, but ειι will not be zero. It can be recovered easily using Eq.(2.9) after the in-plan stresses are obtained.

Figure 2.5. Plane stress problem. The dimension of the solid in the thickness (z) direction is much smaller than that in the x and y directions. All the forces are applied within the x-y plane, and hence the displacements are functions of x and y only.

Figure 2.6. Plane strain problem. The dimension of the solid in the thickness (z) direction is much larger than that in the x and y directions, and the cross-section and the external forces do not vary in the z direction. A cross-section can then be taken as a representative cell, and hence the displacements are functions of x and y only.

Similarly, for the plane strain problems, the stresses σχζ and σγζ are zero, but σzz will not be zero. It can be recovered easily using Eq.(2.9) after the in-plan strains are obtained.

The system equations for 2D solids can be obtained immediately by omitting terms related to the z direction in the system equations for 3D solids. The stress components are

There are three corresponding strain components at any point in 2D solids, which can also be written in a similar vector form

The strain-displacement relationships are

where u, v are the displacement components in the x,y directions, respectively. The strain-displacement relation can also be written in the following matrix form:

where the displacement vector has the form of

and the differential operator matrix is obtained simply by inspection of Eq. (2.26) as