Strain Transformation Equations

10 Key excerpts on "Strain Transformation Equations"

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  eBook - ePub

  Introduction to Composite Materials

  • Hong T. Hahn, Stephen W. Tsai, StephenW. Tsai(Authors)
  • 2018(Publication Date)
  • Routledge

    (Publisher)

  Figure 2.2 , the components of stress and strain will change following prescribed patterns. This variation is called the transformation equation of stress and strain. The state of stress or strain remains the same, independent of the coordinate system, but the magnitude of its components change.

  In conventional materials, physical properties do not change with reference coordinates. This class of materials is isotropic. The transformation of stress or strain has no special meaning or utility with the exception of two special orientations; viz., the principal axes where the shear component vanishes, and the maximum shear orientation which is 45 degrees away from the principal axes. These special orientations are useful for conventional materials because a number of failure theories can be applied using the maximum principal stresses or the maximum shear stresses.

  In composite materials, we need to know the transformation equations of stress and strain for a number of reasons.

  First, the properties of composites are not isotropic. The state of stress or strain existing relative to the on-axis configuration is important in determining the stiffness and strength of composite materials. We can then use the transformation equations, for example, to find the on-axis stress from the applied stress in an off-axis orientation or vice versa.

  Secondly, transformation equations are needed to determine the principal stresses or strains. The same transformation equations also define the invariants of stress and strain. The concepts of principal axes and invariants are fundamental for the understanding of composite materials. The same concepts can be extended to those for stiffness and strength. They will be explained later.

  Finally, transformation equations for stress and strain, together with the on-axis stress-strain relations of Chapter 1 , can be used to determine the off-axis compliance and modulus of unidirectional composites. The sequence of operations is illustrated in Figure 2.3

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  Mechanics of Materials, International Adaptation

  • Timothy A. Philpot, Jeffery S. Thomas(Authors)
  • 2022(Publication Date)
  • Wiley

    (Publisher)

  13

  Strain Transformations

  13.1 INTRODUCTION

  The discussion of strain presented in Chapter 2 was useful in introducing the concept of strain as a measure of deformation. However, it was adequate only for one-directional loading. In many practical situations involving the design of structural or machine components, the configurations and loadings are such that strains occur in two or three directions simultaneously.

  The complete state of strain at an arbitrary point in a body under load can be determined by considering the deformation associated with a small volume of material surrounding the point. For convenience, the volume, termed a strain ­element , is assumed to have the shape of a block. In the undeformed state, the faces of the strain element are oriented perpendicular to the x, y, and z reference axes, as shown in

  Figure 13.1a

  . Since the element is very small, deformations are assumed to be uniform. This assumption means that

  FIGURE 13.1

  1. planes that initially are parallel to each other will remain parallel after ­deformation, and
  2. lines that are straight before deformation will remain straight after deformation, as shown in
     Figure 13.1b
     .

  The final size of the deformed element is determined by the lengths of the three edges dx′, dy′, and dz′. The distorted shape of the element is determined by the angles θ′

  xy

  , θ′

  yz

  , and θ′

  zx

  between faces.

  The Cartesian components of strain at the point can be expressed in terms of the deformations by using the definitions of normal and shear strain presented in Section 2.2

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  Mechanics of Materials

  An Integrated Learning System

  • Timothy A. Philpot, Jeffery S. Thomas(Authors)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  459 CHAPTER 13 Strain Transformations 13.1 Introduction The discussion of strain presented in Chapter 2 was useful in introducing the concept of strain as a measure of deformation. However, it was adequate only for one-directional loading. In many practical situations involving the design of struc- tural or machine components, the configurations and loadings are such that strains occur in two or three directions simultaneously. The complete state of strain at an arbitrary point in a body under load can be determined by considering the deformation associated with a small volume of material surrounding the point. For convenience, the volume, termed a strain element, is assumed to have the shape of a block. In the undeformed state, the faces of the strain element are oriented perpendicular to the x, y, and z reference axes, as shown in Figure 13.1a. Since the element is very small, deformations are assumed to be uniform. This assumption means that a. planes that initially are parallel to each other will remain parallel after de- formation, and b. lines that are straight before deformation will remain straight after defor- mation, as shown in Figure 13.1b. 460 CHAPTER 13 Strain Transformations x y z dx dy dz (a) (b) dz dy dx yz zx xy ′ θ ′ ′ ′ θ ′ ′ θ FIGURE 13.1 The final size of the deformed element is determined by the lengths of the three edges dx ′ , dy ′ , and dz ′ . The distorted shape of the element is determined by the angles θ ′ xy , θ ′ yz , and θ ′ zx between faces. The Cartesian components of strain at the point can be expressed in terms of the deformations by using the definitions of normal and shear strain presented in Section 2.2.

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  Mechanics of Materials

  • Timothy A. Philpot(Author)
  • 2014(Publication Date)
  • Wiley

    (Publisher)

  Table 13.1 Corresponding Variables in Stress and Strain Transformation Equations Stresses Strains  x  x  y  y  xy  xy /2  n  n  nt  nt /2 allow the determination of the shear strain  nt associated with any two orthogonal lines oriented in the n and t directions in the x–y plane. With these equations, the sign conventions used in their development must be rigorously followed: 1. Normal strains that cause elongation are positive, and strains that cause contraction are negative. 2. A positive shear strain decreases the angle between the two lines at the origin of coordinates. 3. Angles measured counterclockwise from the reference x axis are positive. Conversely, angles measured clockwise from the reference x axis are negative. 4. The n–t–z axes have the same order as the x–y–z axes. Both sets of axes form a right- handed coordinate system. x y xy 2    Negative shear strain  xy at origin. y x dx dy O x dx y dy xy 2 —      y x O n t 40° n dn t dt t n O dt dn   nt 2 —    An element of material at point O is subjected to a state of plane strain with strains specified as  x  600 ,  y  300 , and  xy   400 rad. The deflected shape of the element subjected to these strains is shown. Determine the strains acting at point O on an element that is rotated 40 counterclockwise from the original position. Plan the Solution The Strain Transformation Equations will be used to compute  n ,  t , and  nt . SOLUTION The strain transformation equation n x y xy cos sin sin cos 2 2            (13.3) will be used to compute the normal strains  n and  t . Since counterclockwise angles are positive, the angle to be used in this instance is   40.

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  Mechanical Properties of Engineered Materials

  • Wole Soboyejo(Author)
  • 2002(Publication Date)
  • CRC Press

    (Publisher)

  64 Chapter 3 In all the above co-ordinate systems, six independent stress compo-nents are required to fully describe the state of stress on an element. Luckily, most problems in engineering involve simple uniaxial or shear states [Fig. 3.1). Hence, many of the components of the above stress tensors are often equal to zero. This simplifies the computational effort that is needed for the calculation of stresses and strains in many practical problems. Nevertheless, the reader should retain a picture of the generalized state of stress on an element, as we develop the basic concepts of mechanical properties in the subsequent chapters of this book. We will now turn our attention to the basic definitions of strain. 3.3 BASIC DEFINITIONS OF STRAIN Applied loads or displacements result in changes in the dimensions or shape of a solid. For the simple case of a uniaxial displacement of a solid with a uniform cross-sectional area [Fig. 3.3(a)], the axial strain, e , is can be defined simply as the ratio of the change in length, w, to the original length, 1. This is given by [Fig. 3.3(a)]: ^ = u/ (3.8) Note that uniaxial strain is a dimensionless quantity since it represents the ratio of two length terms. Furthermore, strain as described by Eq. (3.8), is often referred to as the engineering strain. It assumes that a uniform displacement occurs across the gauge length [Fig. 3.3(a)]. However, it does not account for the incremental nature of displacement during the deformation process. Nevertheless, the engineering strain is generally satis-factory for most engineering purposes. Similarly, for small displacements, a shear strain, y, can be defined as the angular displacement induced by an applied shear stress. The shear strain, y, is given by y = w /l = tan 0 (3.9) where y, w, / and 6 are shown schematically in Fig. 3.3(b). The angle y has units of radians. However, the shear strain is generally presented as a dimen-sionless quantity.

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  Introduction to Finite Element Analysis Using MATLAB and Abaqus

  • Amar Khennane(Author)
  • 2013(Publication Date)
  • CRC Press

    (Publisher)

  Stress and Strain Analysis 153 5.3.8 T RANSFORMATION OF THE S TRAIN T ENSOR Like the stress tensor, the strain tensor transforms according to the transformation law of second-order tensors. If the components of the strain tensor are known in the basis ( e 1 , e 2 , e 3 ) , then its components in the basis ( e 1 , e 2 , e 3 ) are obtained in both index and matrix notations as km = l ki l mj ij (5.95) [ ] = [ Q ][ ][ Q ] T (5.96) The components l ij or Q ij are the cosines of the angles formed by the unit vectors ( e i , e j ) . The inverse transformations are obtained as ij = l ki l mj km [ ] = [ Q ] T [ ][ Q ] (5.97) 5.3.9 E NGINEERING R EPRESENTATION OF S TRAIN Like the stress tensor, the strain tensor is symmetric and therefore possesses only six independent components. Engineers also prefer to substitute for the shear strains the engineering shear strains as ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 2 3 4 5 6 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ≡ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 11 22 33 γ 12 γ 23 γ 13 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ≡ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ xx yy zz γ xy γ yz γ xz ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ≡ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ xx yy zz 2 xy 2 yz 2 xz ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ (5.98) With this notation, the transformation law for strain in the case of a rotation around the axis 3, or axis z , is written as ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 11 22 33 γ 12 γ 23 γ 13 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ cos 2 ψ sin ψ 0 sin ψ cos ψ 0 0 sin ψ cos 2 ψ 0 − sin ψ cos ψ 0 0 0 0 1 0 0 0 − 2 sin ψ cos ψ 2 sin ψ cos ψ cos 2 ψ − sin 2 ψ 0 0 0 0 0 0 0 cos ψ − sin ψ 0 0 0 0 sin ψ cos ψ ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ × ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 11 22 33 γ 12 γ 23 γ 13 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ (5.99)

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  Design for Thermal Stresses

  • Randall F. Barron, Brian R. Barron(Authors)
  • 2011(Publication Date)
  • Wiley

    (Publisher)

  In the discussion of the stress–strain relationships and temperature field equations in this chapter, we will consider only materials that may be considered homogeneous (the same material throughout) and isotropic (properties the same in all directions). For design purposes, many engineering materials may be treated as homogeneous and isotropic with little error. Materials such as wood, layered composites, and reinforced concrete are exceptions to this statement.

  5.2 Strain Relationships

  The relationships between the displacement of parts of a material and the resulting strain on the material depend only on the geometry or coordinate system used. The forces causing the strain do not enter into the strain expressions, and no material properties appear. Let us look first at the definitions of strain, and then consider the relationships involving the strain components.

  5.2.1 Strain Components

  Let us consider the element in Cartesian coordinates shown in Figure 5.1 . Let us denote the components of the displacement of a particular point (x, y, z ) as u (x -displacement), v (y -displacement), and w (z -displacement). In general, the displacement of a point a small distance from the point in question will be somewhat different.

  Figure 5.1. Differential element used to determine the strains in Cartesian coordinates.

  The extensional strain may be defined as the change in length per unit original length. For example, the extensional strain in the x -direction is

  5.1

  Similarly, the extensional strains in the other two directions may be found from the following relations:

  5.2

  5.3

  The shear strains or distortion strains are defined as the change in the angle between two intersecting planes of the element. Because the strains are small, the tangent of the angle is practically equal to the angle itself. It may be observed that tan ϕ = ϕ, with less than 1 percent error, when ϕ ≤ 0.17  rad = 10°. The shearing strain between the edge of the element in the x -direction and the side of the element in the y

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  Applied Mechanics of Solids

  • Allan F. Bower(Author)
  • 2009(Publication Date)
  • CRC Press

    (Publisher)

  65 3 Constitutive Models: Relations between Stress and Strain The equations listed in Chapter 2 are universal: they apply to all deformable solids. Th ey can-not be solved, however, unless the deformation measure can be related to the internal forces. The constitutive model for a material is a set of equations relating stress to strain (and possibly strain history, strain rate, and other field quantities). Unlike the governing equa-tions in the previous chapter, these equations cannot generally be calculated using fun-damental physical laws (although people are trying to do these calculations). Instead, constitutive models are fit to experimental measurements. Before discussing specific constitutive models, it is helpful to review the basic assump-tions that we take for granted in developing stress-strain laws. They are listed below: A very small sample that is extracted from the solid has uniform properties. When the solid is deformed, initially straight lines in the solid are deformed into smooth curves (with continuous slope), as shown in Figure 3.1. This means that very short line segments (much shorter than the radius of curvature of the curves) are just stretched and rotated by the deformation. Consequently, the • • • Original configuration Deformed configuration e 2 e 1 e 3 FIGURE 3.1 Deformation of an internal material element in a solid. 66 ◾ Applied Mechanics of Solids deformation of a suffi ciently small volume element can be characterized by the defor-mation gradient. The stress at a point in the solid depends only on the change in shape of a vanishingly small volume element surrounding the point. It must therefore be a function of the deformation gradient or a strain measure that is derived from it. If we accept the preceding assumptions, it means that we can measure the relationship between stress and strain by doing an experiment that induces a uniform strain in a suit-able sample of the material.

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  Computational Continuum Mechanics

  • Ahmed A. Shabana(Author)
  • 2008(Publication Date)
  • Cambridge University Press

    (Publisher)

  These two measures are, in general, different because they represent the variations of a vector in different directions. Similar comments apply to the shear strain compo- nents. It is, therefore, important to distinguish between the gradient transformation that results from the change of parameters and the transformation of vectors be- tween different coordinate systems. The transformation of the gradient vectors, as previously stated, changes the coordinate system in which the vectors are defined and does not lead to a change of the strain components, which are defined using dot products. On the other hand, the change of parameters, despite the fact that it is a coordinate transformation, does not change the coordinate system in which the vector and its gradients are defined. This important difference between these two types of transformations was discussed in Chapter 1 of this book, and recog- nizing this difference is important in understanding the physical meaning of the strain components as well as understanding the large deformation finite element formulations discussed in this book. EXAMPLE 2.7 In Example 4, it was shown that the Green–Lagrange strain tensor for a beam whose axis is parallel to the reference X 1 axis and subjected to an elongation d is given by e ¼ 1 2 J T J ÿ I ÿ  ¼ d l d 2l þ 1   0 0 0 " # In this case, e 11 represents the axial normal strain, and all other strain compo- nents including the shear strains are equal to zero. The strain components in another coordinate system " X 1 " X 2 that is rotated by an angle b with respect to the coordinate system X 1 X 2 can be written as " e ¼ A T eA where A is the transformation matrix defined as A ¼ cos b ÿ sin b sin b cos b ! Using this transformation, it can be shown that " e ¼ A T eA ¼ e 11 cos 2 b ÿe 11 sin b cos b ÿe 11 sin b cos b e 11 sin 2 b ! 78 Kinematics

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  Theory and Analysis of Elastic Plates and Shells

  • J. N. Reddy(Author)
  • 2006(Publication Date)
  • CRC Press

    (Publisher)

  Here we consider a special coordinate transformation. Let the principal material coordinates be denoted by ( x 1 , x 2 , x 3 ) and the plate coordinates be ( x, y, z ). Further, assume that the x 1 x 2 -plane is parallel to the xy -plane, but rotated counterclockwise about the z = x 3 -axis by an angle θ (Figure 1.4.1). Conversely, ( x, y, z ) is obtained from ( x 1 , x 2 , x 3 ) by rotating the x 1 x 2 -plane clockwise by an angle θ . Then the coordinates of a material point in the two coordinate systems are related by Eq. (1.2.63). We wish to write the relationships between the components of stress and strain in the two coordinate systems. 1.4.2 Transformation of Stress Components Let σ 11 , σ 12 , · · · , σ 33 denote the stress components in the material coordinates ( x 1 , x 2 , x 3 ) and σ xx , σ xy , · · · , σ zz be the stress components referred to the plate coordinates ( x, y, z ). In the notation of Section 1.2, ( x 1 , x 2 , x 3 ) is the system with bars, ( x, y, z ) is the system without bars, and θ is the angle of rotation about the x 3 = z axis in the counter-clockwise direction. Since stress is a second-order tensor, we have from Eqs. (1.2.73) and (1.2.74) the following relations among the two sets of components: θ 1 x x y θ 3 x = z 2 x Ch 1: Vectors, Tensors, and Equations of Elasticity 33 Figure 1.4.1 A plate with global and material coordinate systems.

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