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ANALYSIS OF STRAIN

Universidad Aut´ onoma del Carmen Facultad de Ingenieria y Tecnolog´ıa – DAIT Dise˜ no mecanico

October 21, 2019

(UNACAR)

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October 21, 2019

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´Indice

1

Deformation

2

Affine Transformations

3

Infinitesimal Affine Deformations

4

A Geometrical Interpretation of the components of Strain

5

Strain Quadratic of Cauchy

6

Principal Strains. Invariants

(UNACAR)

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October 21, 2019

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Contenido

1

Deformation

2

Affine Transformations

3

Infinitesimal Affine Deformations

4

A Geometrical Interpretation of the components of Strain

5

Strain Quadratic of Cauchy

6

Principal Strains. Invariants

(UNACAR)

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October 21, 2019

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Deformation

The change in the relative position of points is a deformation, and the study of deformations is the province of the analysis of strain. A rigid body is an ideal body such that the distance between every pair of its points remains invariant throughout the history of the body. The behavior of rigid bodies subjected to the action of forces is investigated in the mechanics of rigid bodies, where it is shown that the possible displacements in a rigid body consist of translations and rotations. Let the body τ , occupying in the underformed state some region R, be referred to an orthogonal set of Cartesian axes 0 − X1 X2 X3 (Fig.1) fixed in space.

(UNACAR)

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October 21, 2019

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Deformation The coordinates of typical point P of τ in the unstrained state are (x1 , x2 , x3 ). In the strained state the points of τ will occupy some region 0 R , and we denote the coordinates of the same material point P by 0 0 0 (x1 , x2 , x3 ). We shall be concerned only with continuous deformation of R 0 into R and shall write the equations characterizing the deformation in the form, 0 0 0 xi = xi (x1 , x2 , x3 ) = xi (x), (i = 1, 2, 3) (1.1) We shall further suppose that Eqs (1.1) have a single-valued inverse 0

0

0

0

xi = xi (x1 , x2 , x3 ) = xi (x ),

(i = 1, 2, 3)

(1.2)

0

so that the transformation of points from R into R is one-to-one. To ensure the existence of the single-valued inverse, it would suffice to assume 0 that the functions xi (x), are of class C 1 in R and have a non-vanishing Jacobian in the region. We shall assume that this is so. (UNACAR)

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October 21, 2019

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Contenido

1

Deformation

2

Affine Transformations

3

Infinitesimal Affine Deformations

4

A Geometrical Interpretation of the components of Strain

5

Strain Quadratic of Cauchy

6

Principal Strains. Invariants

(UNACAR)

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October 21, 2019

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Affine transformations

The properties of the general linear transformation of points, 0

x1 = α10 + (1 + α11 )x1 + α12 x2 + α13 x3 0

x2 = α20 + α21 x1 + (1 + α22 )x2 + α23 x3 0

x3 = α30 + α31 x1 + α32 x2 + (1 + α33 )x3 Or, written more compactly, 0

x2 = α20 + (δij + αij )xj ,

(UNACAR)

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(i, j = 1, 2, 3),

(2.1)

October 21, 2019

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Affine transformations

where the coefficients αij are constants, are well known. Since it is desirable to demand the existence of an inverse, Eqs. (2.1) must be 0 0 0 solvable for the variables x1 , x2 , x3 as functions of x1 , x2 , x3 . It follows that the determinant |δij + αij | of the coefficients of the unknowns entering into the right-hand member of (2.1) must not vanish. It is obvious that the inverse transformation 0

xi = β10 + (δij + δij )xi ,

(i, j = 1, 2, 3),

(2.2)

Is likewise linear.

(UNACAR)

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October 21, 2019

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Affine transformations It is easy to see from (2.1) and (2.2) that an affine transformation carries planes into planes, and hence a rectilinear segment joining the

Figure: 1

points P 0 (x10 , x20 , x30 ) and P(x1 , x2 , x3 ) is transformed into a rectilinear 0 0 0 0 0 0 segment joming the corresponding points P 0 (x10 , x20 , x30 ) and P (x1 , 0 0 x2 , x3 ) (Fig. 1). (UNACAR)

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October 21, 2019

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Affine transformations This follows from the fact that the rectilinear segment P 0 P can be thought of as joming two points P 0 and P on the intersection of two planes S1 and S2 ; under the transformation (2.1) points P 0 and P go over 0 0 0 0 into points P 0 and P , which lie on the intersection of planes S1 and S2 , into which the planes S1 and S2 are carried by the transformation. We shall denote the unit base vectors, directed along the coordinate axes x1 , x2 and x3 , by e1 , e2 and e3 respectively. Thus, a vector A whose components along the coordinate axe are A1 , A2 , A3 can be written as A = e1 A1 + e2 (A2 + e3 )A3 = ei Ai ,

(i = 1, 2, 3).

Since the vector A = ei Ai is uniquely determined once its components Ai (i = 1, 2, 3) are prescribed, we can represent the vector A by the symbol Ai . Under the transformation (2.1) the vector Ai = xi − xi0 , joining the points P 0 x 0 and P(x), is carried into another vector 0

0

0

Ai = xi − xi0 (UNACAR)

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October 21, 2019

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Affine transformations

0

In general, vectors Ai and Ai differ in direction and magnitude. 0

xi = αi0 + xi + αij xij We have 0

0

0

Ai = xi − xi0 = (αi0 + xi + αij xj ) − (αi0 + xi0 + αij xj0 ) = (xi − xi0 ) + αij (xj − xj0 ) = Ai + αij Aj , Or

0

δAi = Ai − Aj = αij Aj ,

(UNACAR)

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(i, j = 1, 2, 3).

October 21, 2019

(2.3)

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Affine transformations

It is clear from (2.3) that two vectors A and B, whose components are 0 0 equal transform into two vector Ai and Bi whose components are again equal. Also two parallel vectors obviously transform into parallel vectors. Hence, two equal and similarly oriented rectilinear polygons located in different parts of the region R will be transformed into two equal and 0 similarly oriented polygons in the transformed region R .

(UNACAR)

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October 21, 2019

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Affine transformations

Thus, the different parts of the body τ , when the latter is subjected to the transformation (2.1), experience the same deformation independently of the position of the parts of the body. For this reason, the deformation characterized by (2.1) is called a homogeneous deformation. 0 Consider the transformation (2.1), and let the variables xi be subjected to another affine transformation, 00

0

xk = γk0 + (δki + γki )xi .

(2.4)

Recalling the definition of the Kronecker delta, we can write (2.4) as 00

0

0

xk = γk0 + xk + γki xi .

(UNACAR)

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October 21, 2019

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Affine transformations

00

0

Let Ak be the transform of the vector Ak ; then 00

0

0

00

0

0

A00k ≡ xk − xk0 = (γki0 + xk + γki xi )(γk0 + xk0 + γki xi0 ) 0

0

0

0

0

0

= (xk − xk0 ) + γki (xk − xk0 ) = Ak + γki Ai Or

0

00

0

0

δAk = Ak + Ak = γki Ai ,

(UNACAR)

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(i, k = 1, 2, 3).

October 21, 2019

(2.5)

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Affine transformations The product of the two successive affine transformations (2.1) and (2.4) is equivalent to the single transformation obtained by substituting in (2.4) 0 the values of xi in the terms of x, from (2.1). Thus one has xk00 = γk0 + (δki + γki )[αi0 + (δij + αij )xj ] = αk0 + γk0 + (δkj + αkj + γkj )xj + αi0 γki + αij γki xj . Now if the coefficients αij and γij are so small that one is justified in neglecting their products in comparison with the coefficients themselves, then 00

xk = αk0 + γk0 + xk + (αkj + γkj )xj .

(UNACAR)

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October 21, 2019

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Affine transformations

The product transformation likewise carries the point (x10 , x20 , x30 ) to the 00 00 00 point (x10 , x20 , x30 ) where 00

xk0 = αk0 + γk0 + xk0 + (αkj + γkj )xj0 .

(UNACAR)

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October 21, 2019

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Affine transformations The vector Ak = xk − xk0 is thus transformed into the vector 00

00

00

Ak = xk − xk0 = (xk − xk0 ) + (αkj + γkj )(xj − xj0 ) = Ak + (αkj + γkj )Aj Or

00

δAk = Ak − Ak = (αkj + γkj )Aj ,

(j, k = 1, 2, 3).

(2.6)

One of the chief sources of the difficulty that confronts arises from the fact that the principle of superposition of effects and the independence of the order of transformations are no longer valid. A transformation of the type (2.1), in which the coefficients are so small that their products can be neglected in comparison with the linear terms, is called an infinitesimal affine transformation.

(UNACAR)

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October 21, 2019

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Contenido

1

Deformation

2

Affine Transformations

3

Infinitesimal Affine Deformations

4

A Geometrical Interpretation of the components of Strain

5

Strain Quadratic of Cauchy

6

Principal Strains. Invariants

(UNACAR)

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October 21, 2019

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Infinitesimal Affine Deformations In this section we shall be concerned with the problem of separating the infinitesimal affine transformation defined by Eq.(2.3), 0

δi = Ai − Ai = αij Aj ,

(3.1)

into two component transformations: one these corresponds to a rigid body motion; the other, which we have termed pure deformation, will be investigated in detail in the next section. We seek first the conditions on the coefficients αi , if the deformation is to be one of rigid body motion (that is, one consisting of rotation and translation) alone. A rigid body motion is characterized by the fact that the length p A = |A| = Ai Ai . Of any vector A is unchanged by the transformation. If we replace the Ai in this formula by Ai + δAi and denote the change in length A by δA, we get AδA = Ai + δAi (3.2) (UNACAR)

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October 21, 2019

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Infinitesimal Affine Deformations

plus terms of higher order in δAi , which are neglected, since we are concerned with the infinitesimal affine transformation.When the expressions for δAi , given by (3.1) are inserted in (3.2), one finds that AδA = αij Ai Aj or when written out in full, AδA = α11 A21 + α22 A22 + α33 A23 + (α12 + α21 )A1 A2 +(α23 + α32 )A2 A3 + (α31 + α13 )A3 A1

(UNACAR)

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October 21, 2019

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Infinitesimal Affine Deformations

Since for a rigid body transformation δA vanishes for all values of A1 , A2 , A3 , we must have α11 = α22 = α33 = 0 α12 + α21 = α23 + α32 = α31 + α13 = 0 Hence a necessary and sufficient condition that the infinitesimal transformation (3.1) represent a rigid body motion is αij = −αji

(UNACAR)

(i, j = 1, 2, 3).

Presentaci´ on

(3.3)

October 21, 2019

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Infinitesimal Affine Deformations In this case, the set of quantities αij is said to be skew-symmetric. when the coefficients αij are skew-symmetric, the transformation (3.1) takes the form δA1 = −α21 A2 + α13 A3 , δA2 = α21 A1 − α32 A3 , δA3 = −α13 A1 + α32 A2 This transformation can be written as the vector product of the infinitesimal rotation vector ω = ei ωj and the vector A, namely e1 e 3 2 3 ω2 ω3 δA = ω × A = ω1 x1 − x 0 x2 − x 0 x3 − x 0 1 2 3

(UNACAR)

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October 21, 2019

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Infinitesimal Affine Deformations if we take  w1 ≡ α32 = −α23 = 21 (α32 − α23 ), w ≡ α13 = −α31 = 12 (α13 − α31 ),  2 w3 ≡ α21 = −α12 = 12 (α21 − α12 ),

(3.4)

The equations representing the rigid body motion can be obtained by observing that Ai = xi − xi0 , and that 0

0

0

δA1 = Ai − Ai = (xi − xi0 ) − (xi − x10 ) 0

0

= (xi − xi ) − (xi0 − xi0 ) = δxi − δxi0

δxi = δxi0 + δAi = δxi0 + (ωXA)i (UNACAR)

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(1) October 21, 2019

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Infinitesimal Affine Deformations

Then the rigid body portion of the infinitesimal affine transformation (2.1) can be written as  0  δx1 = δx1 δx2 = δx20 + ω3 (x1 − x10 )   δx3 = δx30 − ω2(x1 − x10 ) + ω1 (x2 − x20 )

− ω3 (x2 − x20 ) + ω2 (x3 − x30 ) − ω1(x3 − x30 ) (3.5)

(UNACAR)

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October 21, 2019

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Infinitesimal Affine Deformations

0

The quantities δx30 ≡ xi0 − xi0 are the components of the displacement vector representing the translation of the point P 0 (x 0 ) (see Fig. 1), while the remaining term (3.5) represent rotation about the point P0. At the beginning of the this section, we proposed the problem of separating the infinitesimal affine transformation δA = αij Aj into two component transformations, one of which is to represent rigid body motion alone; we have seen that this rigid body motion correspond to a transformation in which coefficients are skew-symmetric; that is αij . Now any set of quantities αij may be discompose into a symmetric and a skew-symmetric set in one, and only one, way.

(UNACAR)

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October 21, 2019

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Infinitesimal Affine Deformations We can thus write 1 1 αij = (αij + αji ) + (αij − αji ) 2 2 then Eq.(3.1) can be written as 1 1 δAi = αij Aj = [ (αij + αji ) + (αij − αji )]Aj , 2 2 or δAi = eij + Aj ωij Aj ,

(3.6)

where 1 eij = eji ≡ (aij + aji ) 2 1 ωij = −ωji ≡ (aij − aji ), 2 (UNACAR)

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October 21, 2019

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Infinitesimal Affine Deformations The skew-symmetric coefficient ωij correspond to a rigid body motion, and from (3.4) it can be seen that they are connected with the components of rotation,ω1 ω2 ,ω3 , by the relations ω32 = ω1

ω13 = ω2 ,

ω21 = ω3

It is clear from Eqs.(3.6) for the transformation of the components of a vector that an infinitesimal affine transformation of the vector Ai can be descomposed into transformation δAi = ωij Aj , representing rigid body motion, and into transformation δA = eij Aj ,

(3.7)

representing pure deformation. The symmetric coefficients eij are called components of the strain tensor, and they characterize pure deformation. we shall investigate the properties of the strain tensor in the next section. (UNACAR)

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October 21, 2019

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Contenido

1

Deformation

2

Affine Transformations

3

Infinitesimal Affine Deformations

4

A Geometrical Interpretation of the components of Strain

5

Strain Quadratic of Cauchy

6

Principal Strains. Invariants

(UNACAR)

Presentaci´ on

October 21, 2019

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A Geometrical Interpretation of the components of Strain The geometrical significance of the components of strain eij entering into (3.7)can be readily determined by inserting the expressions (3.7) in the formula (3.2), which then takes the form AδA = Ai δAi = eij Ai Aj , or eij Ai Aj δA = A A2

(4.1)

If initially the vector A is parallel to the X1 − axis, so that A = A1 , and A2 = A2 = 0, then it follows from (4.1) that δA = e11 A

(4.2)

Thus, the component e11 of the strain tensor represent the extension, or change in length per unit length, of a vector originally parallel to the a1 − axis. (UNACAR)

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October 21, 2019

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A Geometrical Interpretation of the components of Strain

Figure: 2

(UNACAR)

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October 21, 2019

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A Geometrical Interpretation of the components of Strain

Hence,if all component of the strain tensor with the exception of e11 vanish, then all unit vectors parallel to the x1 − axis will be extended by an amount e11 if the strain component is positive and contracted by the same amount if e11 is negative. In this event, one has homogeneous deformation of material in the direction of the x1 − axis. A cube of material whose edges before deformation are l units long will become a rectangular parallelepiped whose dimensions in the x1 − direction are l(1 + e11 ) units and whose dimensions in the directions of the x2 − and x3 − axes are unaltered.

(UNACAR)

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October 21, 2019

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A Geometrical Interpretation of the components of Strain

A cube of material whose edges before deformation are l units long will become a rectangular parallelepiped whose dimensions in the x1 − direction are l(1 + e11 ) units and whose dimensions in the directions of the x2 − and x3 − axes are unaltered. A similar significance can be described to the components e22 and e33 . In order to interpret geometrically such strain components as e23 , consider two vectors A = e2 A2 and B = e3 b3 (Figure 2),initially directed along the x2 − and x3 − axes respectively. Upon deformation, these vectors become A0 = e1 δA1 + e2 (A2 + δA2 ) + e3 δA3 B 0 = e1 δB1 + e2 δB2 ) + e3 (B3 + δA3 )

(UNACAR)

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October 21, 2019

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A Geometrical Interpretation of the components of Strain

0

0

We denote the angle between A and B by θ and consider the change of 0 0 the scalar product of A and B , have 0

0

0

0

A B cosθ ≡ A · B = δA1 δB1 + (A2 + δA2 )δB2 + (B3 + δB3 )δA3 . = A2 δB2 + B3 δA3

(UNACAR)

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October 21, 2019

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A Geometrical Interpretation of the components of Strain

If neglect the products of the changes in the components of the vectors A and B. To the same approximation, we have 0

A ·B cosθ = 0 0 AB

0

(4.3)

A2 + δB2 + B3 δA3 =q p (δA1 )2 + (A2 + δA2 )2 + (δB2 )2 (δB1 )2 + (δB2 )2 + (B3 + δB3 )2 . = (A2 δB2 + B3 δA3 )(A2 + δA2 )−1 (B3 + δB3 )−1 δB2 δB2 . A2 δB2 + B3 δA3 = = + A2 B3 B3 A2

(UNACAR)

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October 21, 2019

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A Geometrical Interpretation of the components of Strain Since all increments int the components of A and B have been neglected except δA3 and δB2 , the deformation can be represent as shown in Fig.3. If we remember that A1 = A3 = B1 = 0 the Eqs.(3.7) yield δB2 = e23 B3 ,

δA3 = e23 A2 .

(4.4)

From (4.3) we have cosθ = cos(

π − α23 ) = sin 2

α23

δB2 δB3 . = α23 = + B3 B2 or α23 = 2e (UNACAR)

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October 21, 2019

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A Geometrical Interpretation of the components of Strain

Figure: 3

(UNACAR)

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October 21, 2019

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A Geometrical Interpretation of the components of Strain Hence a positive value of 2e23 represent a decrease in the right angle between the vectors A and B, which were initially directed along the positive x2 − axes and x3 − axes. Again, from (4.4) and Fig. 3 we see that δA3 0 0 . = e23 , ∠POP = tan POP = A2 δB2 0 . 0 = e23 , ∠ROR = tan ROR = B3 0

0

Since the angles POP and ROR are equal, it follows that,by rotating the 0 0 0 parallelogram R OP Q through and angle e23 about the orgin, one can obtain the configuration showm in Fig. 4. Obviusly it represents a slide or a shear of the element parallel to the x1 x2 − plane, where the amount of slide is proportional to the distance X3 of the eleent from the x1 x2 − plane. A similar interpretation can obviously be made in regard to the components e1 2 and e31 . (UNACAR)

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October 21, 2019

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A Geometrical Interpretation of the components of Strain

Figure: 4

(UNACAR)

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October 21, 2019

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A Geometrical Interpretation of the components of Strain

It is clear that the areas of the rectangle and the parallelogram in Fig. 4. is equal. Likewise an element if volume originally cubical is deformed into a parallelepiped, and the volumes of the cube and parallelepiped are equal if one disregard the products of the changes in the linear elements. Such deformation is called pure shear. The characterization of strain presented in secs. 3 and 4 is essentially due to cauchy. It should be noted that the strain components eij refer to the chosen set of coordinate axes; if the axes ares changed, the eij will, in general,assume different values.

(UNACAR)

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October 21, 2019

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Contenido

1

Deformation

2

Affine Transformations

3

Infinitesimal Affine Deformations

4

A Geometrical Interpretation of the components of Strain

5

Strain Quadratic of Cauchy

6

Principal Strains. Invariants

(UNACAR)

Presentaci´ on

October 21, 2019

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Strain Quadratic of Cauchy 0

With each point P x of a continuous medium, we shall associate a quadric surface, the quadric of deformation, which enable once to determine the elongation of any vector A = ei (xi − xi0 ) that runs from the point P 0 (x 0 ) to some point P(x). Now if a local system of axes Xi , is introduced with origin at the initial point P0(x0 ) of the vector A and with axes parallel to the space-fixed axes,then formula(4.1) characterizing the extension e = δA/A of A can be written as eA2 = eij xi xj .

(5.1)

we consider the quadratic function 2G (x1 , x2 , x3 ) ≡ eij xi xj

(5.2)

and constrain the end point P(x) of the vector A, as yet unspecified, to lie on the quadric surface 2G (x1 , x2 , x3 ) = ±k 2 (UNACAR)

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(5.3) October 21, 2019

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Strain Quadratic of Cauchy

Where k is any real constant and the sign is chosen so as to make the surface real. Comparison of (5.3) with (5.1) leads to the relation e=±

k2 , A2

(5.4)

and the strain quadric takes the form eij xi xj = ±k2.

(UNACAR)

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(5.5)

October 21, 2019

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Strain Quadratic of Cauchy

We refer the quadric surface of deformation (5.5) to a new coordinate 0 0 0 system x1 , x2 , x3 , obtained from the old by a rotation of axes. Let the 0 directions of the new coordinate axes x1 be specified relative to the old 0 system x1 by the table of direction cosines 0

x1 0 x2 0 x3

x1 ι11 ι21 ι31

x2 ι12 ι22 ι32

x3 ι13 ι23 ι33 0

in which l1 , is the cosine of the angle between the x1 −and the x,−axes.

(UNACAR)

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October 21, 2019

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Strain Quadratic of Cauchy The old and the new coordinates are related by the equations 0

0

0

0

0

0

0

0

0

x1 = l11 x1 + l21 x2 + l31 x3 , x2 = l12 x1 + l22 x2 + l32 x3 , x3 = l13 x1 + l23 x2 + l33 x3 Or, more compactly, x1 = lα1 xα0

(5.6)

It is readily shown that the inverse transformation is of the form 0

xi = II α xα The well-known orthogonality relations between the direction cosines can be written in the form liα ljα = δ1j , (UNACAR)

lδ ilδ j = δ1j

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(5.7) October 21, 2019

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Strain Quadratic of Cauchy 0

When the quadric surface is referred to the x1 coordinate system, a new 0 set of strains e1 , is determined and is replaced by the new equation of the surface,namely, 0 0 0 eij xi xj = ±k 2 . Has a geometrical meaning that is independent of the choice of coordinate system (±k 2 = eA2 ); consequently 0

0

0

0

0

eij , xi xj = e1 , xi xj

(5.8)

In other words, the quadratic form eij xi xj is invariant with respect to an orthogonal transformation of coordinates. Equations (5.6) and (5.8) together yield 0 0 0 0 0 eij lαi lβj x αx β = e αβ, x αxβ ,

(UNACAR)

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October 21, 2019

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Strain Quadratic of Cauchy and since the xα0 are arbitrary, 0

eαβ = lαi lβj eij

(5.9)

Similarly it can be shown that 0

eαβ = lαi lβj eij

(5.10)

A set of quantities eij transforming according to the law is said to represent a Cartesian tensor of rank 2. Differentiating 2G (x1 , x2 , x3 ) = eij , xi , xj and noting from that for a pure deformation δAi = eij , Aj = eij , xi , we find that ∂G = eij xj = δAi . ∂xi

(UNACAR)

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(5.11)

October 21, 2019

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Strain Quadratic of Cauchy

Figure: 5

(UNACAR)

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October 21, 2019

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Strain Quadratic of Cauchy

But ∂G ∂xi are the direction rations of the normal v to the quadric surface at the point (xi ), and it follows that the vector δA is directed along the normal to the plane tangent to the surface eij , xi , xj = ±k 2 . This property of the strain quadric will prove useful in the next section, where we discuss the principal axes of the quadric surface and their significance for the deformation.

(UNACAR)

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October 21, 2019

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Contenido

1

Deformation

2

Affine Transformations

3

Infinitesimal Affine Deformations

4

A Geometrical Interpretation of the components of Strain

5

Strain Quadratic of Cauchy

6

Principal Strains. Invariants

(UNACAR)

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October 21, 2019

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Principal Strains. Invariants We seek now the direction ratios of the lines through (x 0 ) whose orientation is left unchanged by the deformation δAi = eij Aj . If the direction of the vector A is not altered by the strain, the δA and A are parallel and their components are proportional. Therefore δAi = eAi . i It should be noted that e = δA Ai , is the extension of each component of A and is thus the extension of A itself, or = δA/A. Equation (5.1) the shows that the extension e is given by the expression e = eij xi xj /A2 . We return now to Ai = eij Ai , from which it is seen that

eij Ai = eAi = eδA ,

(6.1)

(eij − eδ ij)Aj = 0.

(6.2)

or

(UNACAR)

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October 21, 2019

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Principal Strains. Invariants This set (6.2) of three homogeneous equations in the unknowns Aj possesses a nonvanishing solution if, and only if, the determinant of the coefficients of the Aj is equal to zero; that is, |eij − eδij | = 0, or

(6.3)

e11 − e e12 e13 e21 e22 − e e23 = 0 e31 e32 e33 − e

We prove next that three roots e1 , e2 , e3 of this cubic equation in the elongation e are all real. In this notation, formula (6.1) becomes, for any root e = e1 , 1

1

e1 Aj = ej k Ak

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Principal Strains. Invariants 2

We multiply both side Aj and sum over j, getting 1 2

1 2

e1 Aj Ai = ej k Ak Aj 2

(6.4)

2

Similary , from e2 Aj = ej k Ak we have 1 2

1 2

1 2

1 2

e2 Aj Aj = ej k Aj Ak = ekj Ak Aj = ej k Ak Aj

(6.5)

where j anda k have been interchanged and the symmetry of ejk exploited. Comparsion of (6.4) and (6.5) shows that 1 2

(e1 − e2 )Aj Aj = 0

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(6.6)

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Principal Strains. Invariants Now if we assume tentatively that (6.3) has complex roots, then these can be written e1 = E1 + iE2 ,

e2 = E1 − iE2 ,

e3

where E1 , E2 , e3 , are real. If e2 = E1 − iE2 is substituted for e in (6.2), it 2

will be faund that the resulting solutions Aj ≡ aj − ibj are the complex 1

conjugates of Aj, where the latter are obtained by putting e = e1 = E1 + iE2 . Therefore 1 2

Aj Aj = (aj + ibj )(aj − ibj ) = a12 + a22 + a32 + b12 + b22 + b32 6= 0

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Principal Strains. Invariants Hence it follows from (6.6) that e1 − e2 ≡ 2iE2 = 0, or E2 = 0, and the roots ei are all real. From (6.6) it follows that, if the roots e1 and e2 are distinet, then 1 2

1

2

Ai Ai = A ∗ A = 0, so that the corresponding directions are orthogonal. We have seen that at 1

any point (x 0 ) there are three mutually perpendicular directions A , that 1

are left unaltered by the deformation; consequently the vectors A the 1

1

1

deformed vectors A + δ A and δ A are collinear. But (5.11) shows that δA is normal to the quadric surface(5.5), and therefore the principal directions of strain are also normal to the surface and must be the three principal axes of the quadric eij xi xj = eA2 .

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Principal Strains. Invariants

If some of the principal strains ei are equal, then the associated directions become indeterminate but one can always select three directions that are mutually orthogonal.mIf the quadric surface is a surface of revolution, then 1

one direction A, say , will be directed along the axis of revolution and any 1

two mutually perpendicular vectors lying in the plane normal to A may be taken as the other two principal axes.

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Principal Strains. Invariants We recall that e1 , e2 , e3 are the extensions of vectors along the principal axes, while e11 , e22 , e32 are the extensions of vectors along the coordinate axes. If the coordinate axes x1 are taken along the principal axes of the quadric, then the shear strains e12 , e23 , e31 disappear from the equation of the quadric surface and the latter takes the form e1 x12 + e2 x22 + e3 x32 = ±k 2 The cubic equation (6.3) can be written in the form eij − eδij = −e 3 + ϑ1 e 2 − ϑ2 e + ϑ3 = 0

(6.7)

where ϑ1 , ϑ2 , ϑ3 , are the sums of the products of the roots taken one, two, and three at a time:   ϑ1 = e1 + e2 + e3 = ϑ, ϑ2 = e2 e3 + e3 e1 + e1 e2 , (6.8)   ϑ3 = e1 e2 e3 . (UNACAR)

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Principal Strains. Invariants By expanding the determinant (6.7), we see that these expressions can also be written as  ϑ = e11 + e32 + e33 ,     2 2 2  ϑ2 = e32 e33 + e33 e11 + e11 e32 − e31 − e12 − e23     e  e e e e e   = 22 23 + 11 31 + 11 12 ,  e12 e22 e31 e33 e23 e33 (6.9) 2 2 2  ϑ3 =e11 e22 e33 + 2e12 e23 e31 − e11 e23 − e22 e31 − e33 e12 ,      e11 e12 e31      = e12 e22 e23 .   e31 e23 e33 The expression for ϑ2 and ϑ3 can be written compactly by introducing the generalized Kronecker delta. (UNACAR)

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Principal Strains. Invariants If the subscripts p,q,r... are distinct and if the superscripts i, j, k, . . . are the same set of numbers as the subscripts is defined to be +1 or -1. We can now rewrite the formulas (6.9) in the form  ϑ = eij , (i = 1, 2, 3),     1 ij epi eqj , (i, j, p, q = 1, 2, 3), ϑ2 = δpq (6.10) 2!    1 ϑ3 = δ ijk e e e , (i, j, k, p, q, r = 1, 2, 3) pi qj rk 3! pqr Since the principal strains, that is, the roots e1 , e2 , e3 , of (6.7), have a geometrical meaning that is independent of the choice of coordinate system, it is clear that ϑ, ϑ2 , and ϑ3 are invariant with respect to an orthogonal transformation of coordinates.

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Principal Strains. Invariants

The quantity ϑ has a simple geometrical meaning. Consider as a volume element a rectangular parallelepiped whose edges are parallel to the principal directions of strain, and let the lengths of these edges be ι1 , ι2 , ι3 . Upon deformation, this element becomes again a rectangular parallelepiped but with edges of lengths ι1 (ι + e1 ), ι2 (ι + e2 ), ι3 (ι + e3 ).

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Principal Strains. Invariants

Hence the change ϑ in the volume V of the element is δV = ι1 ι2 ι3 (1 + e1 )(1 + e2 )(1 + e3 ) − ι1 ι2 ι3 = ι1 ι2 ι3 (e1 + e2 + e3 ) plus terms of higher order in ei . Thus e1 + e2 + e3 = ϑ =

δV , V

and the first strain invariant ϑ represents the expansion of a unit volume due to strain produced in the medium. For this reason ϑ is called the cubical dilatation or simply the dilatation.

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