Experimental Mechanics of Solids

By Cesar A. Sciammarella and Federico M. Sciammarella

Experimental solid mechanics is the study of materials to determine their physical properties. This study might include performing a stress analysis or measuring the extent of displacement, shape, strain and stress which a material suffers under controlled conditions. In the last few years there have been remarkable developments in experimental techniques that measure shape, displacement and strains and these sorts of experiments are increasingly conducted using computational techniques.

Experimental Mechanics of Solids is a comprehensive introduction to the topics, technologies and methods of experimental mechanics of solids. It begins by establishing the fundamentals of continuum mechanics, explaining key areas such as the equations used, stresses and strains, and two and three dimensional problems. Having laid down the foundations of the topic, the book then moves on to look at specific techniques and technologies with emphasis on the most recent developments such as optics and image processing. Most of the current computational methods, as well as practical ones, are included to ensure that the book provides information essential to the reader in practical or research applications.

Key features:

• Presents widely used and accepted methodologies that are based on research and development work of the lead author
• Systematically works through the topics and theories of experimental mechanics including detailed treatments of the Moire, Speckle and holographic optical methods
• Includes illustrations and diagrams to illuminate the topic clearly for the reader
• Provides a comprehensive introduction to the topic, and also acts as a quick reference guide

This comprehensive book forms an invaluable resource for graduate students and is also a point of reference for researchers and practitioners in structural and materials engineering.

• Language: English
• Publisher: Wiley
• Release date: Mar 26, 2012
• ISBN: 9781119970095

Book preview

1

Continuum Mechanics – Historical Background

The fundamental problem that faces a structural engineer, civil, mechanical or aeronautical is to make efficient use of the materials at their disposal to create shapes that will perform a certain function with minimum cost and high reliability whenever possible. There are two basic aspects of this process selection of materials, and then selection of shape. Material scientists, on the basis of the demand generated by applications, devote their efforts to creating the best possible materials for a given application. It is up to the designer of the structure or mechanical component to make the best use of these materials by selecting shapes that will simultaneously provide the transfer of forces acting on the structure or component in an efficient, safe and economical fashion. Today, a designer has a variety of tools to achieve these basic goals.

These tools have evolved historically through a heritage that can be traced back to the great builders of structures in 2700 BC Egypt, Greece and Rome, to the builders of cathedrals in the Middle Ages. Throughout the ancient and medieval period structural design was in the hands of master builders, helped by artisan masons and carpenters. During this period there is no evidence that structural theories existed. The design process was based on empirical evidence, founded many times in trial and error procedures done at different scales. The Romans achieved great advances in structural engineering, building structures that are still standing today, like the Pantheon, a masonry semi-spherical vault with a bronze ring to take care of tension stresses in the right place. It took many centuries to arrive at the beginning of a scientific approach to structures. It was the universal genius of the Renaissance Leonardo Da Vinci (1452–1519) one of the first designers that gives us evidence that scientific observations and rigorous analysis formed the basis of his designs. He was also an experimental mechanics pioneer and many of his designs were based on extensive materials testing.

The text that follows will introduce the names of the most outstanding contributors to some of the basic ideas of the mechanics of the continuum that we are going to review in this chapter. The next chapter provides background on those who contributed further in the nineteenth century and early twentieth. In the twentieth century many of the basic ideas were reformulated in a more rigorous and comprehensive mathematical framework. At the same time basic principles were developed to formulate solid mechanics problems in terms of approximate solutions through numerical computation: Finite Element, Boundary Element, Finite Differences.

The birth of the scientific approach to the design of structures can be traced back to Galileo Galilei. In 1638 Galileo published a manuscript entitled Dialogues Relating to Two New Sciences. This book can be considered as the precursor to the discipline Strength of Materials. It includes the first attempt to develop the theory of beams by analyzing the behavior of a cantilever beam. A close successor of Galileo was Robert Hook, curator of experiments at the Royal Society and professor of Geometry at Gresham College, Oxford. In 1676, he introduced his famous Hooke’s law that provided the first scientific understanding of elasticity in materials.

At this point it is necessary to mention the contribution of Sir Isaac Newton, with the first systematic approach to the science of Mechanics with the publication in 1687 of Philosophiae Naturalis Principia Mathematica. There is another important contribution of Newton and Gottfried Leibniz that helped in the development of structural engineering; they established the basis of Calculus, a fundamental mathematical tool in structural analysis.

From the eighteenth century, we must recall Leonard Euler, the mathematician who developed many of the tools that are used today in structural analysis. He, together with Bernoulli, developed the fundamental beam equation around 1750 by introducing the Euler-Bernoulli postulate of the plane sections which remain plane after deformation. Another important contribution of Euler was his developments concerning the phenomenon of buckling.

From the nineteenth century we recognize Thomas Young, English physicist and Foreign Secretary of the Royal Institute. Young introduced the concept of elastic modulus, the Young’s modulus, denoted as E, in 1807. The complete formulation of the basis of the theory of elasticity was done by Simon-Denis Poisson who introduced the concept of what is called today Poisson’s ratio.

Ausgustin-Louis Cauchy (1789–1857) the French mathematician, besides being an outstanding contribution to mathematics was one of the early creators of the field of what we call continuum mechanics, both through the introduction of the concept of stress tensor as well his extensive work on the theory of deformation of the continuum.

Claude-Louis Navier (1785–1836), a French engineer, professor of the Ecole de Ponts et Chaussées in Paris, is considered to be the founder of structural analysis by developing many of the equations required for the solution of structural problems and applying them to the construction of bridges.

Another contributor to the basic equations of the continuum is Gabriel Lamé (1795–1870) French mathematician, professor of physics at L’École Polytechnique and professor of probability at the Sorbonne and member of the French Academy. He made significant contributions to the elasticity theory (the Lamé constants and Lamé equations). He was one of the first authors to publish a book on the theory of elasticity. In 1852 he published Leçons sur la théorie mathématique de l’élasticité des corps solides. Another outstanding contributor to the foundations of the mechanics of solids is the French engineer and mathematician Adhemar-Jean- Claude Barré de Saint Venant (1797–1880). His major contributions were in the field of torsion and the bending of bars and the introduction of his principle that is key to the formulation of the solutions in the continuum. The original statement was published in French by Saint-Venant in 1852. The statement concerning his principle is to be found in Mémoires sur la torsion des prismes. The Saint-Venant’s principle has made it possible to solve elasticity problems with complicated stress distributions, by transforming them into problems that are easier to solve.

G. B. Airy (1801–1892) mathematician and professor of Astronomy at Cambridge, introduced in 1862 the concept of stress function. The idea of stress function was applied by Lamé in his work on thick walled vessels, by Boussineq in his work of contact stresses and by Charles Edward English, professor at the Department of Engineering at Cambridge University who applied the idea of stress functions to the solution of problems of stress concentration (1913). August Edward Hough Love (1863–1940), English Mathematician Professor of Natural Philosophy at Oxford author of many papers on the field of Elasticity, author of, A treatise in the Mathematical Theory of Elasticity, first published in 1892.

Tulio Levi-Civita (1871–1941), professor of Rational Mechanics at the University of Padova. He was one of the outstanding mathematicians of the 19th century. He introduced the idea of tensors and tensor calculus that played a fundamental role in the field of mechanics of solids and in the Theory of Relativity. The contributors to the mechanics of solids includes the names of many outstanding mathematicians and physicists of the nineteenth century: James Clerk Maxwell, H. Herzt, Eugenio Beltrami, John Henri Mitchell, Carlo Alberto Castigliano, Luigi Federico Menabrea.

Let us start with a basic approach to see how these different schools of thought are utilized. Here is the scenario: Given a certain body subjected to given loads and given form of support what are the stresses? In strength of materials (i.e., buckling of columns, late eighteenth century) assumptions are made on how body deformations occur and from that stress distributions are obtained. For this approach intuition and experimental measurements are necessary in order to provide an educated guess of how the body deforms. From deformations strains are obtained and then, by using elastic law, stresses are obtained.

Theory of elasticity, a mathematical model of the behavior of materials subjected to deformations (formalized in the late nineteenth and early twentieth century) has a different approach. In theory of elasticity there is no need to make any assumptions in the way the body deforms. All that is needed to solve the problem is:

1. Certain differential equations; and

2. The postulated boundary conditions for the body.

If the solution meets all the conditions of the theory it is possible to say that an exact solution was achieved. At this stage the following question may be asked: What value does this solution have? If experiments are performed using (experimental mechanics) the solution that was obtained using the theory of elasticity will be in agreement with the experiment within a certain number of significant figures. It should be noted that using the theory of elasticity is more complicated than using the strength of materials approach, but it is worth understanding.

The main reason why the theory of elasticity is worth using is because it yields solutions that would not be possible to get using strength of materials. A very simple example of this concept is the case of bending a beam. Strength of material gives the strain and the stress distribution of a section of a beam but these distributions are the correct answers under special conditions: pure bending and away from the applied load. If we have a beam with a concentrated load the stress distribution in the section where the load is applied will be quite different from that given in strength of materials. In many cases the solution of theory of elasticity agrees with strength of materials solutions, but the understanding that comes from theory of elasticity allows us to have a good grasp of the validity of the solutions. In particular it is possible to know when the solutions can be applied to a particular problem.

Today, numerical techniques (i.e., Finite Element Analysis FEA) are used in almost all applications. A FEA practically provides the solution for any possible problem of the theory of elasticity. One may go so far as to say that FEA is all that is necessary to solve problems. However, it should be mentioned at this point that the ability of numerical analysis to provide the solutions is due to the understanding gained through theory of elasticity and continuum mechanics. Another very important distinction should be made between the solution obtained by theory of elasticity and one that is obtained by a numerical method. The theory of elasticity solution provides the answer for all possible solutions of a given problem. The numerical solution provides the answer for specific dimensions and loads. For example, if one wants to analyze what influence a given variable has on a given problem, this can be done in FE but it will require continual computations for all the range of values of interest of the variable. If one knows the theory of elasticity solution the effect of a variable can be deduced directly from this solution. At this stage of our knowledge the possibility of obtaining solutions directly from the theory of elasticity is limited and hence numerical techniques such as FE allow us to solve numerically any possible problem of the theory of elasticity if we have correct information concerning the boundary conditions and the initial conditions in time if we have dynamic problems.

What follows is a review of the basic concepts upon which the theory of the continuum is built. Continuum mechanics is a branch of classical mechanics. It deals with the analysis of the kinematics and the mechanical behavior of materials modeled as a continuous rather than as an aggregate of discrete particles such as atoms. The French mathematician Augustin Louis Cauchy was the first to formulate this model in the early nineteenth century. The continuum model is not only utilized in mechanics, but also in many branches of physics. It is a very powerful concept that helps in the mathematical modeling of complex problems. A continuum can be continually sub-divided into infinitesimal elements whose properties are those of the bulk material. The continuum hypothesis has at its basis the concepts of a representative volume element. What is a representative volume element? It is an actual volume, with given dimensions. To this volume we can apply continuum mechanics and get results that can be verified by experimental mechanics. It is a concept that depends on scales, for example, when we consider a large structure like a dam, the representative volume may be in the order of centimeters, if we consider a metal the representative volume will be of the order of 10 microns or less. What we measure in experimental mechanics is a certain statistical average of what occurs at the level of the microstructure. This characteristic of the continuum model leads us to ambiguities in language, for example, when we talk of properties at a point of the continuum we are in reality referring to the representative volume that has a definite size.

1.1 Definition of the Concept of Stress

The concept of stress is one of the building blocks of continuum mechanics. The stress vector at a point is defined as a force per unit area as in

(1.1) Numbered Display Equation

where ΔF denotes the force acting on ΔA, this vector depends on the orientation of the surface defined by its normal. This vector is not necessarily normal to the surface.

The stress vector does not characterize the state of stress at a given point of the space in the continuum. The state of stress is characterized by a more complex quantity know as the stress tensor σij. The stress tensor has nine components, of which only six are independent. The stress components are represented in a Cartesian system of coordinates by the stress Cartesian tensor that was originally introduced by Cauchy.

(1.2) Numbered Display Equation

The cube shown in Figure 1.1 represents the stress tensor at a point with its nine components (σij = σji).

Figure 1.1 (a) Elementary cube with stress vectors for the faces of the cube. (b) Components of the stress vectors of the faces. (c) Loaded body showing the elementary cube inside the volume.

ch01fig001.eps

This definition has the ambiguity in language we have pointed out before. Figure 1.1 represents a cube in the continuum, but as we said before ideally it represents a system of three mutually perpendicular planes that go through a point. Each of these planes are defined by their normals, in this case the base vector of an orthogonal Cartesian system x, y, z. At each face of the cube there is a resultant stress vector that we have represented by inline with i = x, y, z. As can be seen these vectors are not perpendicular to the faces of the cube. The components of the stress tensor are the projections of the stress vectors in the direction of the coordinate axis. Mathematically, the tensor is a point function that, according to continuum mechanics, is continuous and has continuous derivatives up to the third order. However, when we want to measure it we need to make the measurement in a finite volume. If the finite volume is too small compared to the representative volume, what we measure will appear to us as a random quantity. The fact that we have talked about measuring a stress tensor is again an ambiguity in language. There is no way to measure stresses directly, we will be able to measure deformations and changes of geometry from which we will compute the values of stresses.

1.2 Transformation of Coordinates

All our measurement procedures will require us to define a coordinate system that we need to specify. But to handle this information in posterior manipulations it may be necessary to switch coordinate systems. A tensor is an entity that mathematically is defined by the way it transforms. In the following derivations we are going to go in an inverse way, define the components and then find out how they transform. It is a classical way through which historically the stress tensor was defined. We consider the equilibrium of a tetrahedron, as in Figure 1.2.

Figure 1.2 Equilibrium of a tetrahedron at point P of a continuum. (a) Component of the stress vectors acting on the different elementary areas. (b) Angular orientation of the rotated axis.

ch01fig002.eps

Introducing an arbitrary oblique plane, where it intersects the three mutually perpendicular reference planes creates a tetrahedron. A tetrahedral element about a point P is defined. The axis x′ of the rotated Cartesian coordinates system is perpendicular to oblique plane whereas y′ and z′ are tangent to the plane orientation of the axis x′ and can be established by the angles shown in Figure 1.2 (b). Areas for the triangular elements formed by the coordinates axis and by the intersection of the oblique plane with the coordinates planes are given by,

(1.3) Numbered Display Equation

Where nx’i are the direction cosines of the normal inline with respect to the coordinate axis. The projection equations of static equilibrium can be applied to get the components shown in Figure 1.2. To utilize the projection equations, the first step is to obtain the summation of forces in the x′ direction. Recall that the force corresponding to each stress is: σ × Aσ. Next it is important to obtain the component of the force in x′ direction. Force due to σx is σx Ax = σx Aonx’x. The component of force in x′ is given as (σx Aonx’x) nx’x.

The same procedure is utilized for the other components and the summation of forces in x′ direc-tion gives,

(1.4)

Numbered Display Equation

For a complete transformation of the stress components with respect to the arbitrary oblique surface, the shear stresses inline and inline must be computed. Directional cosines for y′ and z′ as in x′ are defined as,

(1.5)

Numbered Display Equation

(1.6)

Numbered Display Equation

These equations are sufficient for the determination of the stress components on any internal surface in which an arbitrarily selected tangential set of coordinates is used (y’ z’). For a complete transformation of the stress tensor shown earlier to that of a rectangular element oriented by the x’ y’ z’ coordinate system, the six stresses on the two surfaces with normals in the y’ and z’ must also be determined. The component inline are:

(1.7)

Numbered Display Equation

(1.8)

Numbered Display Equation

(1.9)

Numbered Display Equation

The above equations give all the components of the stress tensor when the Cartesian axis orientation is changed. Although these equations have been derived using a finite tetrahedron the postulation is that these relationships continue to be valid in the limit when the tetrahedron dimensions go to zero and the tetrahedron merges with the point P.

1.3 Stress Tensor Representation

The nine components of σij, with i, j = x, y, z of the stress vectors are the components of a second-order Cartesian tensor called the Cauchy stress tensor, which completely defines the state of stresses at a given point, with the notation inline and is defined as,

(1.10)

Numbered Display Equation

The first index i indicates that the stress acts on a plane normal to the xi axis, and the second index j denotes the direction in which the stress acts. A stress component is positive if it acts in the positive direction of the coordinate axes, and if the plane where it acts has an outward normal vector pointing in the positive coordinate direction. The above notation is a standard notation in continuum mechanics and sometimes the coordinate axis are represented by xi with i = 1, 2, 3. In such a case the components of the stress tensor become σij with i, j = 1, 2, 3. We have derived the expressions of how the stress tensor transforms under a change of the coordinate system; from an xi system to a x’i system. The components σij in the initial system are transformed into the components inline in the new system according to the tensor transformation rule that utilizing matrix notation can be represented by,

(1.11) Numbered Display Equation

In (1.11) R is the rotation matrix and the symbol T indicates the transpose matrix

(1.12)

Numbered Display Equation

The above operation can be accomplished by using MATLAB® matrix routines. In MATLAB® matrices can be entered manually, or by using some pre-defined MATLAB® functions.

1.3.1 Two Dimensional Case

Figure 1.3 represents the stress tensor transformation in 3D. This figure can be simplified if one has a 2D state of stresses. The cube of the 3D space becomes a square in two dimensions and the tetrahedron becomes a triangle. Let us say that the stress tensor is such that: σz = τzx = τzy = 0 the stress tensor becomes,

(1.13) Numbered Display Equation

Figure 1.3 Transformation of the stress tensor.

ch01fig003.eps

Figure 1.4 illustrates the rotation of the stress tensor in two dimensions. The normal defines the corresponding plane where the components of the stress tensor need to be computed. The normal is the outwards normal and the positive rotation is counterclockwise.

Figure 1.4 Rotation of the stress tensor in 2D. The normal indicates the plane where the components are computed, the angle θ defines the rotation. (a), (b), (c) components resulting from each one of the components of the stress tensor.

ch01fig004.eps

The components are given by equations (1.14) to (1.16).

(1.14) Numbered Display Equation

(1.15) Numbered Display Equation

(1.16) Numbered Display Equation

1.4 Principal Stresses

In a 3D state of stress there are three mutually orthogonal planes such that the corresponding stress vectors are normal to the corresponding planes. This means these planes have no shear components. The orientations of the planes are called principal directions (also known as Eigen values of the tensor). The values of the stress vectors are called principal stresses inline .

The principal stresses can be ordered in a way such that inline . In the algebraic sense σ1 is the largest value. It is important when dealing with principal stresses to include the corresponding sign. For example, if the stresses are positive σ1 is the largest in absolute value and σ3 is the smallest. If negative there are two basic cases: If the signs are different for two of the three components the algebraic definition should be upheld example: inline ) where σ3 would be the largest negative value.

The components σij of the stress tensor depend in particular on the coordinate system at the point under consideration. However, the stress tensor is a physical quantity and hence it is independent of the coordinate system chosen to represent it. That is it has a fix position in the 3D space. Therefore there are invariant quantities associated with a stress tensor. The word invariants implies that these quantities are independent of the coordinate system; or saying it in a different way, they have the same values no matter what system of coordinates we select. A stress tensor has three independent invariant quantities associated with it. One set of invariants is the values of the principal stresses of the stress tensor. In mathematics the values of the principal stresses are called Eigen values. The directions of the principal stresses in the space are the second set of invariants. Their direction vectors are the principal directions or eigenvectors. Since the basic property of the principal stresses is the direction of the normal to the face of the plane, we can write,

(1.17) Numbered Display Equation

where inline is a constant of proportionality, and in this particular case corresponds to the magnitude σi of the normal stress vector or principal stress.

1.4.1 How to Calculate Principal Stresses after Making the Transformation

Looking back to the transformation coordinates carried out in Section 1.2 our new axes are defined. It is now necessary and very important to relate this new axis to satisfy the equilibrium condition. Applying the equilibrium conditions, means that for the new axes x’y’ and z’ we must satisfy the conditions,

(1.18) Numbered Display Equation

Calling σi the principal stresses and summing up the forces in the x’, y’ and z’ directions the following equilibrium conditions are obtained:

(1.19) Numbered Display Equation

Recalling that

(1.20) Numbered Display Equation

Since the n’s are the directional cosines, there is a homogeneous system that has three equations with three unknowns (σi and two of the directional cosines). A theorem of algebra tells us that in order to have a solution different from the trivial solution zero, the determinant of the coefficients must be equal to zero.

(1.21a) Numbered Display Equation

Expanding the determinant gives us the so called characteristic equation of the tensor, a cubic equation:

(1.21b)

Numbered Display Equation

The above equation can be written,

(1.22) Numbered Display Equation

Where

(1.23) Numbered Display Equation

(1.24) Numbered Display Equation

(1.25)

Numbered Display Equation

As said before, the principal stresses are unique for a given stress tensor. Hence, it follows from the characteristic equation that I1, I2 and I3, called the first, second, and third stress invariants, are invariants regardless of the particular system of coordinates selected. Since equation (1.22) is a cubic equation, it does not have a closed form solution. The literature presents a number of approaches to the solution of the cubic equation. MATLAB® has routines that can be utilized to compute the solution of the cubic equation. Once principal stresses are determined one can go back to the equations below and solve for the directional cosines.

(1.26) Numbered Display Equation

1.4.2 Maximum and Minimum Shear Stresses

The maximum shear stress is equal to one-half the difference between the largest and smallest principal stresses, and acts on the plane that bisects the angle between the directions of the largest and smallest principal stresses, that is, the plane of the maximum shear stress is oriented 45o from the principal stress planes. The maximum shear stress is expressed as

(1.27) Numbered Display Equation

If, inline then,

(1.28) Numbered Display Equation

The normal stress component acting on the plane of the maximum shear stress

(1.29) Numbered Display Equation

1.5 Principal Stresses in Two Dimensions

The equations derived above become simplified when dealing with a state of stresses in two dimensions. In fact (1.11) becomes,

(1.30) Numbered Display Equation

This equation gives the second degree equation,

(1.31) Numbered Display Equation

The solution of this equation is,

(1.32) Numbered Display Equation

The direction of the principal stresses can be found directly by making the shear stress given by (1.16) equal to zero,

(1.33) Numbered Display Equation

The above equation gives two solutions that represent the two orthogonal principal stresses. Since (1.33) provides two solutions, to know without ambiguity the direction of σ1 it is necessary to compute an additional trigonometric function,

(1.34) Numbered Display Equation

Knowing both tangent and sine it is possible to establish without ambiguity the direction of σ1 because the quadrant of the angle 2θ is defined. The maximum shear is defined by

(1.35) Numbered Display Equation

The angle is given by,

(1.36) Numbered Display Equation

Again to determine the angle without ambiguity,

(1.37) Numbered Display Equation

1.6 The Equations of Equilibrium

In the previous developments the concept of a stress tensor and the associated transformations are considered. Those concepts correspond to properties that are defined at a point. Now the emphasis shifts to what happens between two neighboring points. This way the equations of equilibrium can be derived. These equations are partial differential equations that involve the components of the stress tensor. These equations are required to have a stress function that satisfies the equilibrium of the continuum, see Figure 1.5.

Figure 1.5 Equilibrium of the elementary cube at a point of the continuum.

ch01fig005.eps

The cube represents neighbor planes in the continuum. One set of planes has the components of the stress tensor. The other plane contains stress tensor components of a neighboring point. This is a mathematical model that defines the behavior of the continuum. This model is in agreement with all the experimental determinations. Towards the middle of the last century other definitions have been introduced but their applicability is reduced to some very special media. To analyze the equilibrium we must introduce forces per unit of volume, F (for example weight, or centrifugal force).

In one set of planes (for x-dir) we have the components σx, inline and inline . The next plane has the increments of these components. By definition the increments are given as

unnumbered Display Equation

Summing the components in the x direction gives (where Fx is body force − weight).

(1.38)

Numbered Display Equation

Simplifying previous equation results in

(1.39) Numbered Display Equation

This is also true for y and z directions

(1.40) Numbered Display Equation

Recall that inline is a force per unit volume. In particular, what if I have a solution that provides the components of the stress tensor? These components must satisfy the previous equations because if they do not it means the solution is incorrect. The forces projection equations of statics are satisfied. What about equilibrium of the moments? The fact that the stress tensor is symmetric (for example inline ensures the validity of the moment equilibrium equations.)

The condition of equilibrium with respect to the centroid of cube requires that, inline , Figure 1.6. Similar relationships can be derived for all the other shear components. Then in general

(1.41) Numbered Display Equation

Figure 1.6 Moment with respect to O is equal to zero.

ch01fig006.eps

1.7 Strain Tensor

In the previous derivations we defined the stress tensor. Parallel to the forces in continuum mechanics we need to develop geometry of deformations that correspond to those forces. In continuum mechanics, there is a general theory of deformation. This theory has to satisfy a metric of the Cartesian space that requires that the distances between points are given by the sum of the squares of the components. This leads to non linear strain tensors. These tensors complicate the solution of problems of the mechanics of the continuum because they transform the system of equations into non linear systems. The developers of continuum mechanics quickly realized this difficulty and proceeded to create the small deformation theory, or also small displacement theory. Of course the basis of the adoption of this theory was the fact that deformations of structural materials are small quantities compared to 1. Consequently, this theory deals with infinitesimal deformations of the continuum. By an infinitesimal deformation it is meant that the displacements ||u||<<1 and the displacement gradients are small compared to unity, inline making feasible the linearization of the Lagrangian finite strain tensor, that is, all the second order terms are removed. The resulting linearized tensor violates one of the basic requirements of the definition of strain, it is not invariant upon rigid body motions. Hence, even if the deformations are small, but rotations are important, the tensor will give non zero strain components. This aspect is very important in the case of methods that measure displacements such us moiré, and speckle methods. With these basic concepts underlying the infinitesimal deformation theory, strain and a point is defined,

(1.42) Numbered Display Equation

This equation deals with the change of segment length.

Besides the change of length the geometry of deformation includes the change of the angle made by two segments, see Figure 1.7.

Figure 1.7 The equation should be mentioned after change of the angle made by two segments.

ch01fig007.eps

The derivation of the linearized strain tensor can be explained with the help of an infinitesimal geometry drawing Figure 1.8. We consider an infinitesimal rectangular material element with dimensions dx, dy (Figure 1.8), which after deformation, takes the form of a rhombus.

Figure 1.8 Analysis of the deformation of the continuum at the neighborhood of a point.

ch01fig008.eps

An element of area is represented by a square with size inline . A deformation is applied to the medium causing the element to change position and shape. The displacements of the points in the medium are represented by two continuous functions of x and y. They are written as

(1.43) Numbered Display Equation

It is possible to define the displacement in the neighborhood of a point, as the strain change at which a point was defined.

(1.44) Numbered Display Equation

A similar expression can be applied for the y direction

(1.45) Numbered Display Equation

The previous equations are utilized to define the deformation experienced by the segments Δx and Δy as the medium is deformed. Utilizing our definition of strain given initially

(1.46) Numbered Display Equation

As explained before this definition leads to a non linearity in the strain tensor due to the fact that Q′D′ has to be computed as the sum of the square of the two components given in the figure. In view of the difficulties that arise when this occurs the earlier developers of continuum mechanics simplified the relationship by replacing Q′D′ by its projection on the x axis. Practically it means that the angle alpha shown in the figure has to be a small angle such that cos α ∼ 1. This means that the equations that we are going to derive are limited to small deformations and small rotations. Therefore, if we want to compute the deformations of a steel component that experiences small deformations the above equations can be used. For the deformation of car tires these equations will not work! The strain tensor components resulting from these simplifications as stated before are called the linearized strain tensors. The simplification introduced produces the following equation

(1.47) Numbered Display Equation

Similar derivation for the y direction yields

(1.48) Numbered Display Equation

The shear component can be computed utilizing a similar simplification

(1.49)

Numbered Display Equation

Since the angles should be small the tangent is equal to the arc inline

(1.50) Numbered Display Equation

Defining w as the displacement in the z direction we can generalize the above quantities

(1.51) Numbered Display Equation

Utilizing the derivations just made we can now define the strain tensor at a given point of a continuum medium

(1.52) Numbered Display Equation

Where inline

The tensor defined here is linearized and has the same transformation equations as the stress tensor. The tensor will have three mutually orthogonal directions that define the principal directions and along these directions act the principal strains, inline . All the derivations of tensorial properties that we have tried concerning the stress tensor apply also to the strain tensor.

1.8 Stress – Strain Relations

An experimental relationship between stresses and strains is incorporated into a logical framework of mechanics to produce formulas for the analysis and design of structural members. If a material model does not fit experimental data well, we get errors in theoretical predictions. Yet a model that can fit the experimental data very well may be so complex that no analytical model can be built. The choice of material model is dictated both by the experimental data and by accuracy needs of analysis.

The stress and strain tensors are related to each other. The constitutive equations provide the relationship between the two. While the mechanics of continuum mathematically models the physical reality, constitutive equations need to be based on experiments that allow the corresponding properties of real materials to be derived.

Each strain is dependent on each stress

(1.53)

Numbered Display Equation

This must be repeated for the other eight terms (81 – constants). Since shear normally is symmetric this reduces to six terms each (36 – constants). This is known as the compliance matrix – that at least 36 material constants are required to describe the most general linear relationship between stresses and strains. Recalling that C12=C21, C13=C31 (shown that W1 = W2) where the w’s are energies, reducing independent constants to 21.

There are important simplifications to constitutive equations. Homogeneous – elastic properties do not change from point to point. Isotropy, regardless of direction in space shows the properties are the same. Many engineering materials are not isotropic so for these the 21 independent material constants must be used. Isotropic materials only require two independent material constants to describe its linear inline relationship. How to classify depends on: (1) material properties with orientation, (2) scale we are looking at and (3) type of information that is desired.

1.8.1 Homogeneous or Not?

At the atomic, crystalline or grain size level materials are non homogeneous. Depending on what type of information is required crystalline bodies can be grouped into classes for the purpose of defining the independent material constants needed in the linear stress-strain relationship. A few classifications of material groups will be analyzed. If a material has one plane of symmetry (xy), there can be no interaction between the out of plane shear stresses and remaining strains.

(1.54)

Numbered Display Equation

Orthotropic materials have two orthogonal planes of symmetry. If we rotate a sample by 90° about the x or y axis we will obtain the same inline relation

(1.55)

Numbered Display Equation

For orthotropic materials the normal strains are not affected by the shear stresses, and the shear strains are not affected by the normal stresses. This is not true for general anisotropic materials.

1.8.2 Material Coordinate System

If the coordinate system is transformed from the xyz coordinate system (1.54) and (1.55) will change. These equations are valid for a specific coordinate system. This system is known as the material coordinate system. Looking at composite materials as in Figure 1.9, that are made out of a matrix reinforced with continuous fibers, the material properties will change with the orientation of the fibers.

Figure 1.9 Looking at composite materials, that is a material made out of a matrix reinforced with continuous fibers, the material properties will change with the orientation of the fibers.

ch01fig009.eps

Fibers are inherently stiffer and stronger than the bulk material, due to the reduction of defects and alignment of crystals along the fiber axis. It is clear that the mechanical properties will be different in the direction of the fiber and perpendicular to the fiber. If the properties of fiber and epoxy are averaged each lamina can be regarded as an orthotropic material and the directions parallel and perpendicular to the fiber are the material axis directions. Stacking the laminate with different fiber orientations creates a composite laminate. Overall the properties can be controlled by orientation of fibers and stacking sequence – for certain stacking sequences laminate will respond like an orthotropic material. If the properties of the orthotropic material are identical in all three directions the material is said to have a cubic structure (FCC, BCC). Below are definitions that provide the relationship between the laminates.

(1.56)

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If isotropic you only need constants E and G or E and ν where G = E/2(1 + ν)

(1.57)

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1.8.3 Linear, Elastic, Isotropic Materials. Lamé Constants

The utilization of Lamé constants is an alternative way to express the stress-strain relationship, the following equation is an example expressed in indicial notation where x, y and z are replaced by 1, 2 and 3. In the equation that follows when an index variable appears twice in a single term it implies that we are summing over all of its possible values.

(1.58) Numbered Display Equation

Where i, j = 1, 2, 3

Where:

(1.59) Numbered Display Equation

and

(1.60) Numbered Display Equation

The above expression is an invariant of the strain tensor, the sum of the normal strains. The following equations are the relation between the Lamé constants and the Young’s modulus and Poisson’s ratio.

(1.61) Numbered Display Equation

(1.62) Numbered Display Equation

1.9 Equations of Compatibility

In continuum mechanics, a compatible strain tensor field in a deformed body is a field that is produced by a continuous, single-valued, displacement field. One can prove that such a field is unique. The equations of compatibility are mathematical expressions that provide the conditions under which the strain field satisfy the conditions of continuity and uniqueness. These equations were first derived for linear elasticity by Barré de Saint-Venant in 1864. Later Beltrami in 1886 provided a more general expression. An intuitive understanding of the equations can be gained by imagining the continuum made of infinitesimal volumes. Each volume is deformed but in doing so it must be connected to all the neighbor elements without gaps or overlaps. The equations of compatibility are the mathematical conditions that must be satisfied to guarantee that in the process of deformation gaps or overlaps are not introduced. The adjective compatible applied to a deformation indicates that the basic postulate of the continuum is satisfied. The mathematical expression of compatibility conditions depends on the utilized tensor field. If the tensor is non-linear the mathematical conditions of compatibility are quite involved and are called the Mainardi-Codazzi conditions of compatibility. If the tensor is linearized the expression of the equations become much more simple.

When working through a problem it is possible to define either displacements or stresses as unknowns, or a combination of the two. A problem arises if stresses are chosen as unknowns. The stresses may give displacements that violate the basic assumption of continuum mechanics. The solution may result in the presence of discontinuities inside the continuum, for example voids or overlaps, as previously explained. To avoid this problem, the strains computed after the stress-strain relationship are applied must satisfy a system of partial equations of the second order. These equations are called equations of compatibility, also called the Beltrami equations. Using the indicial notation, and indicating the second derivatives with two indices following a semicolon

(1.63) Numbered Display Equation

Expanding (1.63) one obtains,

(1.64) Numbered Display Equation

If the solution is done using displacements as unknowns the compatibility equations will automatically be satisfied. If we choose solutions that involve linear stresses the compatibility equations will also be automatically satisfied because the compatibility is second order equations.

References

The following publications are provided for more information on the topics presented in this chapter.

1. Nair, S. (2000) Introduction to Continuum Mechanics, Cambridge.

2. Mase, G.E. (1969) Theory and Problems of Continuum Mechanics, Schaum’s Outline Series, McGraw Hill.

3. Boresi, A.P. and Chong, K.P. (2000) Elasticity in Engineering Mechanics, 2nd edn, John Wiley & Sons, Inc., NY.

4. Liu, I.-S. (2002) Continuum Mechanics, Springer.

5. Timoshenko, S. History of Strength of Materials, Dover Paper Back, February 1, 1983.

6. Todhunter, I. (1960) History of the Theory of Elasticity and of the Strength of Materials from Galilei to Lord Kelvin Two Volumes in Three Parts, Dover.

2

Theoretical Stress Analysis – Basic Formulation of Continuum Mechanics. Theory of Elasticity

2.1 Introduction

Theory of elasticity is a branch of a more general theory known as continuum mechanics. Theory of elasticity was the first theory to evolve to analyze the behavior of solids subjected to loads. Why? The linear relationship between stress and strain simplifies the solutions. As we have theory of elasticity we can have theories to analyze the behavior of loaded bodies that have constitutive equations of different types: theory of plasticity, theory of viscoelasticity, and so on. These theories will differ from theory of elasticity in the constitutive equations. However, other equations besides the constitutive equations, the so called field equations, will also be present in the other theories.

2.2 Fundamental Assumptions

Basic assumptions of the classical theory are:

1. Displacements and strains are small.

2. The material is homogenous isotropic.

3. There is a linear relationship between stresses and strains.

One important aspect of the classical theory is the fact that the obtained solutions are given in the original geometry of the body, Lagrangian description. When one deals with fairly rigid structures there is not much difference between the un-deformed and the deformed structure. If the structure is very flexible but elastic, (i.e., tire) the problem will still be a problem of the theory of elasticity, but will be a non linear problem and the constitutive equations can be different from those utilized in the classical theory.

2.3 General Problem

Theory of elasticity has been created to solve the problem of bodies under given conditions of load and supports (connection with other bodies). We have the following unknowns.

(2.1) Numbered Display Equation

(2.2) Numbered Display Equation

(2.3) Numbered Display Equation

There are six components of the stress tensor, six components of the strain tensor, three components of the displacement vector. The total number of unknowns adds up to 15. In order to solve a problem such as the one posed before there must be an equal number of equations and unknowns. So … what are the equations at our disposal?

(2.4) Numbered Display Equation

(2.5) Numbered Display Equation

(2.6) Numbered Display Equation

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There are three equations of equilibrium (these are partial differential equations). There are also six equations defining the stress strain relationship. Finally, there are six equations defining the strain displacement relationships. This provides us 15 equations to work with; therefore we can solve the problem.

You may be asking, what about the compatibility equations? Let us recall that the continuity equations are used when the solutions are formulated in terms of the stresses only. The strains are computed as a function of the stresses. The strains must satisfy the compatibility equations.

Some difficulties arise since we have to solve partial differential equations hence it is necessary to know the branch of mathematics from which we can get solutions to partial differential equations. For this reason very often the theory of elasticity is called mathematical theory of elasticity. Since there is no given preset procedure to solve partial differential equations, solutions in the past were worked out using special methods by many different authors.

Today, the availability of numerical techniques such as finite elements or boundary value methods provides numerical solutions that are quite general and make it feasible to solve any kind of problem in the theory of elasticity. These solutions of course are based on approximate methods and hence the solutions have a certain degree of error. This is a complex subject and there is a methodology available to extract the best solutions possible depending on the particular problem that is being solved.

There is another point to be taken into account. There is one more difference between the solution using the theory of elasticity and using a numerical method. A theory of elasticity solution provides the answer for all possible cases of a given problem. The numerical solution provides the answer for specific dimensions and loads. Of course one can generate many numerical solutions as a function of given parameters. There are other types of solutions that you will be used to handling, including the strength of materials solution. In strength of materials one assumes the deformation of a body and from this assumption one obtains the stress distribution. In this process one uses intuition and experimental measurements to provide an educated guess of how the body deforms. In theory of elasticity it is not necessary to make any assumptions in the way the body deforms, it is enough to satisfy certain differential equations and the postulated boundary conditions for the body. If the solution meets all the conditions of the theory then it is said there is an exact solution.

What value does this solution have? If we perform experiments using experimental mechanics we will find that the solution will be in agreement with the experiment within a certain number of significant figures. The theory of elasticity is more complicated than the strength of materials approach, but it is worth understanding. You may ask, why is this important? The reason is that theory of elasticity yields solutions that would not be possible to get using strength of materials. In many cases the solution of theory of elasticity agrees with strength of materials solutions, but the understanding that comes from theory of elasticity allows us to have a good grasp of the validity of the solutions.

2.3.1 Boundary Conditions

The solution of partial differential equations requires the definition of the problem. The particular solution depends on:

1. The geometry of the body.

2. The way that the body is loaded; and

3. The way in which it is connected to other bodies that provide support to the analyzed component.

Therefore information on the Boundary Conditions of the problem must be provided. There are two types of classical of boundary conditions:

1. Boundary conditions given by forces.

2. Boundary conditions given by geometrical constraints.

There is a third type of boundary condition which adds some difficulty to the solution of the problem. It is when we have mixed boundary conditions, some components are forces and some are geometrical constraints that are specified at the same points. To simplify the presentation of this topic we will use a 2D example, but the same analysis can be applied to 3D bodies.

Figure 2.1 Boundary conditions in a 2D body.

ch02fig001.eps

In Figure 2.1 there is a body that is fixed in the region A B C and in D C there is an applied distributed load. This example shows the two typical conditions that are present in the boundaries of bodies. Either displacements are prescribed or forces are prescribed. In some cases a boundary can be a mixed boundary in which both displacements and forces are prescribed. In this example no displacement can take place in A B C so displacement = 0. In the rest of the body from C D E A forces are prescribed, from C to D constant load is applied. From D E A no external load is applied (free boundary). At the boundary where the forces are prescribed (CDEA) the internal stresses σx and σy and τxy must be in equilibrium with the applied forces. Thus the previous (transformation) equations must be satisfied at the boundary.

(2.7) Numbered Display Equation

(2.8) Numbered Display Equation

In the region D E A the boundary is free and hence the stresses σB and τB must be equal to zero; then equations (2.7) and (2.8) are equal to zero at the boundary. For C D and DEA we have the following boundary conditions:

Unnumbered Display Equation

Since σB = σx and τB = τxy along CD the top equation shows that σx = −p and τxy = 0 within the body along edge CD. Along the boundary ABC the displacements are equal to zero and the boundary conditions along the points of this segment are:

Unnumbered Display Equation

A case of mixed boundary conditions will be as follows. Suppose that along ABC the same body of Figure 2.1 is supported by a rough surface. The body is constrained to move along the surface ABC (geometrical constraint). At the same time there is a force along the surface of the body T = f N, f is the coefficient of friction and N is the normal reaction at the considered point. At the same time the points of the segment are constrained to move along the boundary. Along the same boundary we have a condition in displacements and in forces.

Returning to the solution of the problem in two dimensions, one has to satisfy the equations of equilibrium which are given as:

(2.9) Numbered Display Equation

(2.10) Numbered Display Equation

The stress-strain relationships are:

(2.11) Numbered Display Equation

The strain-displacement relationships are:

(2.12) Numbered Display Equation

If the solution is in terms of the displacements one has to obtain two functions u(x,y) and v(x,y) that satisfy the previous equations and the specified boundary conditions.

2.4 St. Venant’s Principle

Why is it called a principle? Because there is no formal proof for validity although by experience we know it is valid. There have been many attempts to derive this principle from the basics of mechanics of continuum (no success yet to create a proof). This is a very important principle in the practical sense, because it provides a bridge between many solutions obtained in strength of materials and theory of elasticity. If a system of forces acting on a small region in elastic body is replaced by another force system, acting within the same region and having the same resultant forces and moment as the first system (statically equivalent system of forces), then the stresses at a distance of about twice the size of the application region of the forces will be identical.

In many cases the strength of materials solution is identical to the theory of elasticity except in the region where the loads are applied. In that case theory of elasticity must be used. Take Figure 2.2 and the bar and assume we are pulling in tension (rectangular section of h and b) at a distance 2h the stresses are uniform independent of how forces are distributed.

Figure 2.2 Example of application of St. Venant principle.

ch02fig002.eps

The stress in the bar is given by

(2.13) Numbered Display Equation

Actually the above solution is a solution from the theory of elasticity. Looking at Figure 2.3, equa-tion (2.13) can be applied to the cross-section of the bars away from the region of transition of the cross-sections (central regions of the bars). In the regions of transition there is a complex stress field that needs to be computed by the theory of elasticity. In Figure 2.4 there is another example explaining this principle. It is a beam with a region subjected to the pure bending where the equation of strength of materials is given as,

(2.14) Numbered Display Equation

Where M is the bending moment and y the distance of the section from the neutral axis to the fiber under consideration and Ic the moment of inertia of the cross section of the beam. Equation (2.14) is a solution of the theory of elasticity. However, the photoelastic pattern shows that the solution is valid a certain distance from where the transition of the section of the beam takes place.

Figure 2.3 Bars with uniform cross-section axially load, photoelastic patterns. From M.M. Frocht, Photoelasticty Volume I, Copyright © 1957 by John Wiley & Sons, Inc. Reprinted by permission of John Wiley & Sons, Inc.

ch02fig003.eps

Finally Figure 2.5 shows a beam subjected to bending produced by a concentrated force in the middle of the span. Nowhere in the beam span is the strength of materials solution valid. The basic assumptions of strength of materials to derive the bending equation are not satisfied. The St. Venant principle cannot be applied to this problem.

Figure 2.4 Beam under pure bending (photoelastic pattern). From M.M. Frocht, Photoelasticity Volume I, Copyright © 1957 by John Wiley & Sons, Inc. Reprinted by permission of John Wiley & Sons, Inc.

ch02fig004.epsch02fig005.eps

2.5 Plane Stress, Plane Strain

Although the 3D approach fits all possible cases, simplifying approaches are introduced to reduce the required effort to get solutions. There are two typical approaches: Plane Stress and Plane Strain.

Figure 2.5 Rectangular beam subjected to bending by a concentrated load in the middle of the span. From M.M. Frocht, Photoelasticity Volume I, Copyright © 1957 by John Wiley & Sons, Inc. Reprinted by permission of John Wiley & Sons, Inc.

ch02fig006.eps

Figure 2.6 Figure illustrating the definition of plane stress.

ch02fig007.eps

As shown in Figure 2.6 two parallel planes at a distance t is small with respect to the other two dimensions and limited by a lateral closed surface which constitutes a plate. If the applied loads are parallel to the planes that limit the plate and are constant through the thickness we have the configuration corresponding to a plane stress problem. Calling z the coordinate perpendicular to the limiting planes, the basic assumption of plane stress is σz = τxz = τyz = 0. It can be shown that this assumption does not satisfy the equations of compatibility. It follows that plane stress solutions are not exact solutions of the theory of elasticity. From the practical point of view we can say that these stresses and the corresponding strains are very small with respect to the in-plane stresses and strains. Therefore they can be neglected. In the plane strain problem we have the opposite condition. There is a very long body, a cylinder of arbitrary directrix shape, loaded by symmetrical loads.

The loads are perpendicular to the lateral surface. Imagine a very long tube, for example loaded with internal pressure as seen in Figure 2.7.

Figure 2.7 Illustration of plane strain on a cylinder.

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2.5.1 Solutions of Problems of 2D Using the Airy’s Stress Function

There is a way to solve the two dimensional problems of 2D theory of elasticity by using the Airy stress function. Airy was an Australian Astronomer working in England and proposed this method in 1863. Calling inline the stress function, the equations below relate inline to the stresses.

(2.15) Numbered Display Equation

where inline is a function of x and y that is continuous and has 2nd order derivatives that are also continuous. Substituting into the equilibrium equations

(2.16) Numbered Display Equation

it is possible that the equations of equilibrium are satisfied. However, since this solution is in terms of stresses it must be verified that the compatibility equations are also satisfied. Replacing (2.15) into the compatibility equation

(2.17) Numbered Display Equation

Replacing the strains as a function of the stresses in (2.17),

(2.18) Numbered Display Equation

One gets,

(2.19)

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Substituting inline on the right side, taking into consideration the equations of equilibrium gives the compatibility equation in terms of stress.

(2.20)

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This is known as the compatibility equation in terms of stresses. Finally substituting σx,σy utilizing the first two equations of (2.15) in (2.20), produces the biharmonic equation,

(2.21) Numbered Display Equation

Then the solution of the problems of two dimensional theory of elasticity are reduced to simply satisfying the biharmonic equation when the body forces are constant or zero. In other words all 2D problems in theory of elasticity are solutions of the biharmonic equation with certain limitations that are consequences of the connectivity of the medium. If the medium is simply connected, that is if a closed curve in the medium can be reduced to a point without crossing any boundary the above statement is correct. If the medium is not simply connected some additional requirements are needed.

2.6 Plane Stress Solution of a Simply Supported Beam with a Uniform Load

The problem is a classical one and can be solved with the help of Airy's polynomials. In this example it will become clear why the St. Venant principle is needed in the approximate solutions. The solution of a simple beam being considered in this section is a complex problem. The beam is a 3D body and then the actual problem is a problem of 3D theory of elasticity and as such is very complex. However some simplifying assumptions can be made and then the solution becomes amenable to being solved by utilizing Airy's polynomials.

This problem will be solved as a 2D plane stress problem. Hence the stresses that are defined as σz = τxz = τyz = 0. It has already been mentioned that plane stress solutions are approximate since the equations of compatibility are not satisfied. What is the meaning of the above statement? The assumption that stresses σz = 0, τxz = 0, τyz = 0 is incorrect; these stresses will be present in the beam. However it can be said that from the application point of view these stresses will be small compared to the other components of the stress tensor. The other interesting aspect of this example is that the St. Venant principle will be used repeatedly so that the problem can be formulated. The price paid by introducing this simplification is that the stress distribution in some critical sections of the beam will not be known. For example at the support where the actual stresses are far from being negligible the distribution may be very high. This leads us to a practice that is common in structural problems. Sections of components of structures such as beams are computed by