Strain Tensor at a Point - Term Paper Example

Strain Tensor at a Point Name: Institutional Affiliation: Date: Table of Contents i.Abstract 3 1.0.Introduction 4 2.0.Deformation 5 2.1.Deformation Gradient 6 3.0.One Dimensional strain 8 4.0.The concept of actual strain 10 5.0.Definition of actual strain 11 6.0.Infinitesimal strain tensor theory 12 7.0.Characteristics of infinitesimal strain tensor 14 8.0.The relation of strain transformation 17 9.0.The compatibility equation for infinitesimal strains 18 10.0.Principal strains 22 11.0.Rules of handling second order strain tensor 23 120.0. Conclusion 24 i. Abstract Second order tensor has a foundational basis and practical application in the fields of biomechanics, physical and earth sciences and engineering. Second order tensor represents the Strain at a point and all the fundamental rules applied in handling second order tensor can be used to handle strain at a point even though there are some differences in the kinematics of stress tensor that makes it even harder to control. Examples of second-order tensors are velocity gradients in fluid flow and stresses and strains in solids. A vivid analysis and visual representation of second-order tensor enhances the understanding and the interpretation of the data on tensor and therefore is of supreme significance to the scientists. This paper aims to give a clear view in the representation of strain at a point by the second order tensor. More specifically the article will focus primarily on opening the definition of strain in one-dimensional problems. Development of infinitesimal strain tensor and its relation to the customary one-dimensional stress measures used in material testing and elementary strength of materials. Clearly differentiate between general deformation, strain, and rotations in infinitesimal and significant strain cases. Show the independence of the stress tensor from rotations. Discuss various rules of handling second order tensors that may be used to manage the stress tensor. Explain the reasons for the need for other pressure measures even though for small strains. Derive the Strain compatibility equations and show the difficulties in recovering displacement from stress filed among other areas. 1.0. Introduction Because of their relation to the deformation and failure of materials, strains and stresses are of enormous importance to scientists working in biomechanics, engineering, and earth science fields. The matrices involved in strains are very complex a fact that makes tensor of such nature very hard to inference especially when the data is of significant volume. The expectation is that because of the dichotomy between strains and stresses and because of the dichotomy between the relative displacement and the forces of contact, the underlying meaning of stress tensor would present a simple task to formulate an unfortunate assertion (Hashiguchi &Yamakawa, 2012). In kinematics, stresses can induce deformation on materials since materials in the real sense are not boundlessly stiff. The measure of distortion is a strain. Integrating stress through some field of action results to displacement, which gives the measurement of the movement of the components of the material (Feng, Shi, Zhang & Teng, 2016). The outcome of this is the alteration of the size and shape of the body. In short, stresses bring about strains which lead to displacement which in turn produces a change in the size and shape of the material. Strain theory is an original principle concept in continuum and structural mechanics. The field of displacement and strain may be measured directly using a function that assigns a non-negative size to all vectors in the movement space (Iliopoulos & Michopoulos, 2013). A material or body that is capable of stretching and is especially able to regain shape and size when force is released, that is a stretchable body becomes misshaped when a load is applied. This change in length and the relative direction as a result of deformation is referred to loosely as strain, a concept that was developed and introduced by Truesdell and Toupin (1960) in their celebrated work the principles of classical mechanics and field theory (Maugin, 2015). There is the need to find a mathematical expression that describes the displacement the material undergoes upon application of a load or force. In general, the explanation of the strain is that of three dimensions since the body sizes and shapes of materials possess three-dimensional properties. The best part of the paper discusses much of a strain of the deformation of the second-order dimension known as plane strain, where the pressure is described wholly by the alterations in the sizes and shapes in one arrangement of the plain through the material, assuming that no deformation takes place normal to the plane. Even though plane strain has found wide application in the analysis of deformation, the use of its idea and rule in not justified primarily in the natural rock deformation. The geometry of deformations of the second dimension is much easier to comprehend and can be generalized into three size distortion though that presents some difficulty. The two reasons make it of a significant requirement to deeply look into the features of a two-dimensional train. One dimensional strain will be discussed first for its critical application the fields of engineering in to determine the strength and the average resistance of the materials. 2.0. Deformation When describing the deformation of a body completely, it is important to make consideration of the body in its deformed state and its original state; that is before deformation or displacement takes place. The deformation of the body is not uniformly distributed but varies from one part to the other. When the body undergoes deformation, a small line element PQ experiences an alteration from its original length. This distortion affects all the lines stretched in all the possible directions at that particular point. This way a strain at a point comprises two infinite sets as A measurement of the transformation in a linear dimension in all the directions at that particular point and a measure of angular proportional change for every pair of courses at that particular point. The deformation that takes place at a point can have a connection with the neighborhood displacement at the point. 2.1. Deformation Gradient The deformation gradient is the derivative of every part of the deformed vector on all the components of the reference vector. Then Define the displacement of any point as Hence In tensor notation, it is written as Example Y X Considering the figure above where the object is transformed from square the new position shown in the figure. The equations to do this are And the corresponding deformation gradient is The object has been rotated and stretched. But it is the deformation that contributes to stress. It is thus necessary to divide the two mechanics from to determine the strain and stress state. The following steps lead to that. First is a 50% stretch in the x-direction followed by a 25% compression in the y-direction. Then Second step: rotate the configuration to the final x. Thus In matrix form, And The deformation gradient can be written as a product of two matrices: a symmetric matrix and a rotation matrix A practical example of deformation can occur in metals and polymers considered viscoelastic. The creep response can be modeled by Kelvin-Voigt model. The deformation strain is given by the following functional analysis integral 3.0. One Dimensional strain For one-dimensional strain consider a slender rod fixed at one end as in figure 1 below. This bar is the un-stretched or primary, reference, initial, un-deformed figure and the strain is zero in this case. Fixed end Figure 1 This diagram shows pressure applied at point p of the stretched rod; PQ is a line element in the un-stretched rod, P'Q' is the same line element in the extended bar. This kind of deformation can be described in some ways as will be seen later. At this stage, we consider discussing the application of the engineering strain which is probably the simplest stress measure. In determining the strain at point P, in figure 1 above, a small line element PQ emerging from point P in the un-stretched rod will be considered to be low but finite distance denoted as Δx. This distance is referred to as gage length when determining strains experimentally. Pulling the bar moves it to the stretched or deformed position. The points P and Q move to P' and Q' after extending the rod. The displacements are and respectively. When mathematically determining the strain at point Q, take the limit of the mean over Δx as the distance Δx tends to zero as follows: The same approach can be applied when determining tensions at other points. This association that exists between displacements and strains at the point is the strain displacement equation. The equals gives a formula that can be used as a direct calculation of pressures by differentiation in case the variation in the movement is available or can be determined numerically, for instance from measurements taken in experiments. Deforming a line element by stretching it so that it is twice its original length, then the strain is equal to 1. Before deformation, the strain is zero. Changing the shape of a body by shortening its length by half is -0.5. The measurement of pressure is in percentage, for instance, a 100% strain is 1, a 200% strain is a strain of 2 and so on. In practice, most materials used in engineering like concrete and metals experience a minuscule percentage of pressure while materials like rubber can experience strains of the large magnitude of up to 100%. When determining the pressure at point Q, apply the same approach by considering the line PQ to be emanating from point Q. Note: With the availability of the strains at all the points in the rod, it is not easy to ascertain the exact position of the rod in space since the pressures do not include all the information about the possibility of the rigid movement of the body. Hence it is important to get more information if you are to know the exact position of the rod in space 4.0. The concept of actual strain The magnitude of pressure deformation is measurable in several possible ways. Examples of such applicable methods include Euler and the Green-Lagrange strain. Regarding figure 1 above, these strains are expressed as follows; Green –Lagrange Euler –Almasi strain Most of the measurements of strain are applicable in theories of the behavior of materials that are more advanced, especially when the stretches or the deformations are of big magnitude. Besides the detailed discussion of the engineering strain, the right strain will be considered owing to its frequent use to describe testing of material (Hashiguchi, Yamakawa, 2012). 5.0. Definition of actual strain Define a small increment in pressure to the change in length, divided by the original length L: dε=dL/L. Uniformly stretching the rod changes the original of the bar continuously, and the total strain is the accumulation of these increments expressed as The additive nature of actual strain is another reason that makes exact stress in useful. For instance stretching the rod in two steps from lengths L1 to L2 to L3, then the total strain is given by . Which is an implication that the deformation had taken place in a just a single step; this concept does not apply in the engineering strain. The expression below shows the connection between the design and the actual strain; . One significant result of the relation is that the more you reduce the magnitude of the stretch the more, the two strains become similar to each other. Example: A metal wire is 2.5 mm diameter and 2 m long. A force of 12 N is applied to it and it stretches 0.3 mm. Assuming the material is elastic. Determine: i. The stress in the wire ii. The strain in the wire Solution A= One practical example would be to press a tennis ball with the thumbs. The ball undergoes temporary deformation but not easily since there is some internal resistance offered against the force applied by the thumb. The resistance offered is stress. The deformation it has undergone is strain 6.0. Infinitesimal strain tensor theory Infinitesimal strain theory as used in continuum mechanics can be defined as the attempt made to describe the action of deforming a solid material body mathematically. Where the displacement of the particles of the material body is under deformation is assumed to be smaller than even to the slightest extent the relevant aspect of the body (Xiao & Yue, 2014). The assumption, in this case, is that the geometric and constitution characteristics of the body like the measure of the amount of matter contained in the body and the measure of rigidity at each point of space during deformation does not undergo deformation. This assumption is in sharp contradiction with the conventional one-dimensional strain theory. Where the assumption made is the opposite. This assumption makes it easy to simplify the equations of the continuum mechanics. The little strain theory has found a wide range of application in the practical fields of the civil and mechanical engineering to analyze the ability of a structure to withstand stress. These structures are composed of rigid materials such as steel and concrete, because the principal aim of designing using structures using such materials is to lower their stretch or deformation when subjected to some force or workload (Neto, Perić, Owen, 2008). For instance, whenever a heavy loaded vehicle runs over a bridge for example, with certainty we can say that the car exerts some amount of force on the bridge. Nevertheless, it is assumable that the bridge stays unchanged since the transformation or change in shape if any is not relevant considering the total hugeness of the bridge. Infinitesimally small deformations of a body can be described in two different ways: first by basing an approach of linearizing Lagrangian Strain tensor or Euler strain tensor of the finite one-dimensional strain and secondly is from a geometric description. From figure 1, assuming that the stretch and the gradient of displacement are less than unity, that is ǁ x ǁ˂˂1 and ǁ Δx ǁ˂˂1, a geometric linearization of the finite strain tensor theories proceeds as below: Lagrangian strain tensor And the Euler strain tensor: This linearization is an implication that the description by Lagrange and the description by Euler are approximately similar since the difference that exists between the coordinates of a point the material at a space and the material is little. Thus the displacement gradient of the material and the parallel gradient of the material are approximately equal. The following follows Lagrange strain tensor=Euler strain tensor= or Where are the components of the infinitesimal strain tensor. With the assumption given above, it possible to calculate the change in the length per unit length across all the directions from the point only if the tensor ε is given. It is also feasible to calculate the angular change between any two pairs of perpendicular directions at that point and can thus be applied as strain tensor referred to as infinitesimal or small strain tensor. The tensor ε can, therefore, be used as a stress tensor only if the displacement is small. 7.0. Characteristics of infinitesimal strain tensor The expression of the engineering strain of an infinitesimal strain tensor aligned with a vector of length unity m for small strains is; The infinitesimal strain tensor has a close relation to the strain matrix. To derive this relationship, we consider being a plane with the lines, running parallel to x and the line, running parallel with y and finally, running parallel to z. Now, denote the characters in the immense strain tensor in this foundation by. Then For a general strain tensor, the strain characters along the diagonal are referred to as common or direct strains. The other diagonal components, are called shear strains, which is in some instances termed the engineering shear strain and has an association with the general stress tensor by a factor 2. That is . Another important feature of the infinitesimal strains worth considering is the volumetric infinitesimal strain expressed as trace () ,, and the deviatory strains which are defined by . The volumetric strain is the measurement of the changes in volume. For infinitesimal strains, it is related to the Jacobean of transformation slope by. Remember Example: The following data gives displacement field in a body (using the finite element method): In Determine the strains at point A(x=4, y=5 in) Solution: Note that and The quantity is the train component parallel to the normal unit . is mostly referred to as the unit elongation along the , even as is the unit elongation in the direction of the x- coordinate in the x-y plane. These measures of strain can be generalized resulting into a three-dimensional case, and the six shearing strains and the three common strains are as follows: , , , These normal pressures give the measure of the size change of the line elements along the given directions of the coordinates while the six shear strains measure the variations in the shape of the line items that were originally along the particular coordinate directions. Their outcome is the most necessary needed in the characterization of the nature of tension at any particular point in the body. The understanding of the extension and the shearing of the line elements about the XYZ directions is confident that the knowledge gives information enough in determining the shear and elongation of the line in any given direction. The combination of the three strains and the six shear strains looks almost similarly to the primary characteristics of the strain tensor. When grouped as follows, the strains are called stress tensors, and each of these measurements of the characteristics of strain tensor is fundamental to the original strain. Note: in the definition of the geometry of translation, the assumption undertaken is that of small size deformation, in exact terms the derivatives of the strain are of small magnitude. If the distortion produced by the loads is of large scale this theory should not be used; instead, a better accurate method should be developed. 8.0. The relation of strain transformation The connections that network the components of strain in one original frame to any other frame are correspondent to the equations of important change. When we consider those elements in the line that were originally along the x, y and z coordinate direction; the definitions of these nine characteristics of the strain tensor are in the previous equation. The next task is determining the application of the line components that happen to be at an angle with the XYZ coordinates. In a more similar term, what happens if another parallel direction, say x' y' z' is considered. The results obtained in this particular arrangement relating with a different arrangement of nine strain components is shown below: The strain components on any other coordinate components could be obtained by the following expression when the elements to one coordinate component are known: 9.0. The compatibility equation for infinitesimal strains The relationships between strain and displacement can sometimes be inverted when it is necessary, that is, given the field of strain, to calculate the deformation. At this part of the paper, the procedure of doing this is outlined considering the particular case of little strains. For the small change the relationship between strain and displacement is given by: Now the determination of the three displacement components is considered when the six strain components are provided. Note that it is not possible to fully recover the field of displacement that has produced a particular area of strain. The stiff movements do not create any strains. Thus the movement is determined if and only if more information is available apart from the strains, which will give one the information on the quantity of the rotation of the material and the amount of translation that has taken place. Nevertheless, integration of the strain field obtains the area of displacement covered by the motion of the rigid solid body. There is therefore of secondary importance need to ensure that the relations of the field of strain displacement can be integrated (Seo, Woo, 2013). The strain is a symmetrical second order tensor fields, though the vice versa is not true, that is, not every balanced second order strain tensor field can be strain field. The relation of strain displacement gives rise to the frame of scalar differential equations for the three displacement elements. . To make the field of strain be able to be integrated, the compatibility conditions must be satisfied by the strains, an expression of this can be given by the following expression: In an equivalent manner Again, The poof that all the strain fields must satisfy all the conditions above is trivial. The proof is possible by just substituting the strains in the components of the displacement; the equation obtained thus satisfies the following equation which is a clear evidence of the relations The same approach is applicable for the other equations producing a satisfactory result. It is significant to note that for the two-dimensional strain tensor problems where , there is a trivial satisfaction of all the compatibility equations apart from the first equation: It is also of paramount importance to show that: in case the compatibility equations are not satisfied by the strains, then it is not possible to integrate the displacement vector from the strains. In case the equations of the compatibility are met by the strains, and the solid body does not have in it any hole that runs all the way across its thickness, then it is possible to integrate the vectors of displacement from the strains. If the material does not possess the whole connection property, it is feasible to calculate the vector of movement, even though the value may not be single, that is, there is a possibility of obtaining differing solutions depending on the manner in which the path of integration circulates the hole. Referring now to the problem addressed by the paper at this section, that is, method of computing the displacement given the strains. The process is quite trivial, and the illustration in the following example can be used to explain: Example: Consider a two-dimensional strain tensor field also known as plane stress with young's modulus E and poisons ratio v loaded at one end by a force p. The beam has a cross-section with height 2a and a thickness b. The beams strain field is , The first step is finding out if the strain is compatible which requires that the following equation is satisfied for the two-dimensional strain tensor problems, in this case, it is satisfied: For the two dimensional problem and such that the following holds , and From the first two equations we obtain , the first equation can be integrated with respect to and integrating the second equation with respect to to obtain , Where is a function of and is a function of both of which will be found. The functions are calculated by substituting and formulae into the expression for the shear strain as follows: This can be re-written as The first term enclosed in the brackets is function of while the second term enclosed in the bracket is the function of. And because the terms on the left-hand side must reduce to zero for all the values of and, this implies that Where: = an arbitrary constant. The expressions can now be integrated as following Where, c and d are arbitrary constants. The field of displacement can finally be estimated s 10.0. Principal strains Rotating the displacement field components into proper coordinates whereby the strains that cause the shear of the body become equals to zero puts the strain tensor into a diagonal matrix. The strain tensor contains six components which are independent of each other. When the is to be determined the three principal values of the strain tensor must be specified as well as defining the three most significant strain directions, such that you still need to consider using the six parameters in describing the states of the strains (Brady, Brow, 1985). There is a mutual orthogonally relation exhibited by the principal directions as shown in the expression that follows. Therefore because it needs to have two angles to give a specification on the axis, it is mandatory for the second axis to lie in the plane perpendicular making the need for just one aspect in addition. The right-hand side rule and orthogonality can be used to determine the third axis in a unique manner. In the small strain limit the following expression holds Extensions of the lowest and the highest magnitudes take place when du and dx are parallel i.e. Or Where is a constant From the previous equation, when there is no rotation taking place, then the relative displacement of the points that lie parallel to each other can be generalized into the following equation equating this equation with gives rise to the next equation This can be rearranged to form. Therefore the principal strains in form the eigenvalues of the matrix while the represents the eigenvectors. 11.0. Rules of handling second order strain tensor The deformation and displacement of the body under various forces follow some rules that include but not limited to the law of conservation, the constitutive law The normal movements of a body under the forces that cause its deformation are ruled by the law of conservation which gives a deeper insight on the concept of momentum and mass conservation when the body is deformed, Roark, et al. (2002). There are some core relations that need to be brought forward for the clarity of the law. The theorem known as the divergence theorem clears the relation between the components of the vectors as they diverge over some surface that is enclosed with the integral of the volume of the material as it diverges through the displacement within the boundaries. In summary the various variables that are used in the relation and compatibility equations are: the displacement components of the variables are six in numbers; strains are six in number; and the density totaling to nine variables (Khraishi, Shen, 2015). There are also some six kinematics equations that give the relationship between strain and displacement namely the equilibrium equations which three in number and the continuity equations of single nature which are also three in number. 120.0. Conclusion The study of strains is not just for class and theoretical studies but has an extensive application in the field of mechanical and civil engineering and is also applicable in earth sciences. These applications make the study of strains an interesting topic. And still presents a point of research globally for the clarity and the improvement of the theories that are in existence thus far. Second order tensor has a foundational basis and practical application in the fields of biomechanics, physical and earth sciences and engineering. Second order tensor represents the Strain at a point. And all the fundamental rules that are usable in handling second order tensor can be used to handle strain at a point even though there are some differences in the kinematics of strain tensor that makes it even harder to handle. This, therefore, leads to the truth that more research should be done and discoveries made that would simplify the complexities that exist in the kinematics of strain tensor. The geometry of deformations of the second dimension is much easier to comprehend and can be generalized into three-dimension deformation though that presents some difficulty. The two reasons make it of a significant requirement to deeply look into the features of a two-dimensional train. Reference Ayadi, S., Hadj-Taïeb, E., & Pluvinage, G. (January 01, 2007). The numerical solution of strain wave propagation in elastical helical spring =: Beer, F. P. (2011). Mechanics of materials. New York: McGraw-Hill. Chichester, West Sussex, UK: Wiley. Cooper, R. C., Lee, C., Marianetti, C. A., Wei, X., Hone, J., & Kysar, J. W. (2013). Nonlinear elastic behavior of two-dimensional molybdenum disulfide. Physical Review B, 87(3), 035423. Hashiguchi, K., & Yamakawa, Y. (2012). Introduction to finite strain theory for continuum elasto-plasticity. Chichester, West Sussek, U.K: Wiley. Khraishi, T. A., & Shen, Y. -L. (2015). Continuum mechanics: Constitutive equations and applications. Fung, Y. C. (2013). Biomechanics: mechanical properties of living tissues. Springer Science & Business Media. Guzʹ, A. N. (2013). Fundamentals of the three-dimensional theory of stability of deformable bodies. Springer Science & Business Media. Feng, G., Shi, W., Zhang, H., & Teng, T. (2016, August). Based on continuum mechanics thrust rod fatigue life prediction. In Mechatronics and Automation (ICMA), 2016 IEEE International Conference on (pp. 356-360). IEEE.Neto, E. A. S., Perić, D., & Owen, D. R. J. (2008). Computational methods for plasticity: Theory and applications. Chichester, West Sussex, UK: Wiley. Roark, R. J., Young, W. C., & Budynas, R. G. (2002). Roark's formulas for stress and strain. New York: McGraw-Hill. Seo, J. K., & Woo, E. J. (2013). Nonlinear inverse problems in imaging. Chichester, West Sussex, UK: Wiley. Shames, I. H., & Cozzarelli, F. A. (1997). Elastic and inelastic stress analysis. Washington, DC: Taylor and Francis. Boyle, J. T., & Spence, J. (2013). Stress analysis for creep. Elsevier. Muskhelishvili, N. I. (2013). Some basic problems of the mathematical theory of elasticity. Springer Science & Business Media. Maugin, G. A. (2015). Book Review: Encyclopedia of Thermal Stresses, edited by Richard B. Hetnarski: LXXXIII (11 vols., 6643 pp., 3269 illus., 1141 illus. in color), Springer, Dordrecht, Holland, 2014. ISBN 978-94-007-2738-0. Journal of Thermal Stresses, 38(2), 271-275. Iliopoulos, A., & Michopoulos, J. G. (2013). Direct strain tensor approximation for full‐field strain measurement methods. International Journal for Numerical Methods in Engineering, 95(4), 313-330. Xiao, H., & Yue, Z. (2014). Fracture mechanics in layered and graded solids: Analysis using boundary element methods. Read More

In general, the explanation of the strain is that of three dimensions since the body sizes and shapes of materials possess three-dimensional properties. The best part of the paper discusses much of a strain of the deformation of the second-order dimension known as plane strain, where the pressure is described wholly by the alterations in the sizes and shapes in one arrangement of the plain through the material, assuming that no deformation takes place normal to the plane. Even though plane strain has found wide application in the analysis of deformation, the use of its idea and rule in not justified primarily in the natural rock deformation.

The geometry of deformations of the second dimension is much easier to comprehend and can be generalized into three size distortion though that presents some difficulty. The two reasons make it of a significant requirement to deeply look into the features of a two-dimensional train. One dimensional strain will be discussed first for its critical application the fields of engineering in to determine the strength and the average resistance of the materials. 2.0. Deformation When describing the deformation of a body completely, it is important to make consideration of the body in its deformed state and its original state; that is before deformation or displacement takes place.

The deformation of the body is not uniformly distributed but varies from one part to the other. When the body undergoes deformation, a small line element PQ experiences an alteration from its original length. This distortion affects all the lines stretched in all the possible directions at that particular point. This way a strain at a point comprises two infinite sets as A measurement of the transformation in a linear dimension in all the directions at that particular point and a measure of angular proportional change for every pair of courses at that particular point.

The deformation that takes place at a point can have a connection with the neighborhood displacement at the point. 2.1. Deformation Gradient The deformation gradient is the derivative of every part of the deformed vector on all the components of the reference vector. Then Define the displacement of any point as Hence In tensor notation, it is written as Example Y X Considering the figure above where the object is transformed from square the new position shown in the figure. The equations to do this are And the corresponding deformation gradient is The object has been rotated and stretched.

But it is the deformation that contributes to stress. It is thus necessary to divide the two mechanics from to determine the strain and stress state. The following steps lead to that. First is a 50% stretch in the x-direction followed by a 25% compression in the y-direction. Then Second step: rotate the configuration to the final x. Thus In matrix form, And The deformation gradient can be written as a product of two matrices: a symmetric matrix and a rotation matrix A practical example of deformation can occur in metals and polymers considered viscoelastic.

The creep response can be modeled by Kelvin-Voigt model. The deformation strain is given by the following functional analysis integral 3.0. One Dimensional strain For one-dimensional strain consider a slender rod fixed at one end as in figure 1 below. This bar is the un-stretched or primary, reference, initial, un-deformed figure and the strain is zero in this case. Fixed end Figure 1 This diagram shows pressure applied at point p of the stretched rod; PQ is a line element in the un-stretched rod, P'Q' is the same line element in the extended bar.

This kind of deformation can be described in some ways as will be seen later. At this stage, we consider discussing the application of the engineering strain which is probably the simplest stress measure. In determining the strain at point P, in figure 1 above, a small line element PQ emerging from point P in the un-stretched rod will be considered to be low but finite distance denoted as Δx.

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