Review of Force, Stress, and Strain Tensors

Transcription

1 Review of Fore, Stress, and Strain Tensors.1 The Fore Vetor Fores an be grouped into two broad ategories: surfae fores and body fores. Surfae fores are those that at over a surfae (as the name implies), and result from diret physial ontat between two bodies. In ontrast, body fores are those that at at a distane, and do not result from diret physial ontat of one body with another. The fore of gravity is the most ommon type of body fore. In this hapter we are primarily onerned with surfae fores, the effets of body fores (suh as the weight of a struture) will be ignored. A fore is a three-dimensional (3D) vetor. A fore is defined by a magnitude and a line of ation. In SI units, the magnitude of a fore is expressed in Newtons, abbreviated N, whereas in English units the magnitude of a fore is expressed in pounds-fore, abbreviated lbf. A fore vetor F ating at a point P and referened to a right-handed x y oordinate system is shown in Figure.1. Components of F ating parallel to the x y oordinate axes, F x, F y, and F, respetively, are also shown in the figure. The algebrai sign of eah fore omponent is defined in aordane with the algebraially positive diretion of the orresponding oordinate axis. All fore omponents shown in Figure.1 are algebraially positive, as eah omponent points in the orresponding positive oordinate diretion. The reader is likely to have enountered several different ways of expressing fore vetors in a mathematial sense. Three methods will be desribed here. The first is alled vetor notation, and involves the use of unit vetors. Unit vetors parallel to the x-, y-, and -oordinate axes are typially labeled î, ĵ, and ˆk, respetively, and by definition have a magnitude equal to unity. A fore vetor F is written in vetor notation as follows: F = F iˆ + F ˆj + F kˆ (.1) x y The magnitude of the fore is given by F = F + F + F x y (.) 41 K13483_C00.indd 41 9/13/01 11:04:14 AM

2 4 Strutural Analysis of Polymeri Composite Materials +x +y F F x F y P F + Figure.1 A fore vetor F ating at point P. Fore omponents F x, F y, and F ating parallel to the x y oordinate axes, respetively, are also shown. A seond method of defining a fore vetor is through the use of indiial notation. In this ase, a subsript is used to denote individual omponents of the vetoral quantity: F = ( Fx, Fy, F ) The subsript denotes the oordinate diretion of eah fore omponent. One of the advantages of indiial notation is that it allows a shorthand notation to be used, as follows: F = F, where i = x, y, or (.3) i Note that a range has been expliitly speified for the subsript i in Equation.3. That is, it is expliitly stated that the subsript i may take on values of x, y, or. Usually, however, the range of a subsript(s) is not stated expliitly but rather is implied. For example, Equation.3 is normally written simply as F = where it is understood that the subsript i takes on values of x, y, and. The third approah is alled matrix notation. In this ase, individual omponents of the fore vetor are listed within braes in the form of a olumn array: F i Fx F = Fy F (.4) K13483_C00.indd 4 9/13/01 11:04:16 AM

3 Review of Fore, Stress, and Strain Tensors 43 Indiial notation is sometimes ombined with matrix notation as follows: F = { F i } (.5). Transformation of a Fore Vetor One of the most ommon requirements in the study of mehanis is the need to desribe a vetor in more than one oordinate system. For example, suppose all omponents of a fore vetor F i are known in one oordinate system (the x y oordinate system, say) and it is desired to express this fore vetor in a seond oordinate system (the x y oordinate system, say). To desribe the fore vetor in the new oordinate system, we must alulate the omponents of the fore parallel to the x, y, and axes that is, we must alulate F x, F y, and F. The proess of relating fore omponents in one oordinate system to those in another oordinate system is alled transformation of the fore vetor. This terminology is perhaps unfortunate, in the sense that the fore vetor itself is not transformed but rather our desription of the fore vetor transforms as we hange from one oordinate system to another. It an be shown [1,] that the fore omponents in the x y oordinate system (F x, F y, and F ) are related to the omponents in x y oordinate system (F x, F y, F ) aording to: F = F + F + F x x x x x y y x F = F + F + F y y x x y y y y (.6a) F = F + F + F x x y y The terms i j that appear in Equation.6a are alled diretion osines, and are to equal the osine of the angle between the axes of the new and original oordinate systems. An angle of rotation is defined from the original x y oordinate system to the new x y oordinate system. The algebrai sign of the angle of rotation is defined in aordane with the right-hand rule. Equation.6a an be suintly written using the summation onvention as follows: F = F (.6b) i i j j Alternatively, these three equations an be written using matrix notation as Fx Fy = F x x x y x y x y y y x y Fx Fy F (.6) K13483_C00.indd 43 9/13/01 11:04:16 AM

4 44 Strutural Analysis of Polymeri Composite Materials Note that although values of individual fore omponents vary as we hange from one oordinate system to another, the magnitude of the fore vetor (given by Equation.) does not. The magnitude is independent of the oordinate system used, and is alled an invariant of the fore tensor. Diretion osines relate unit vetors in the new and old oordinate systems. For example, a unit vetor direted along the x -axis (i.e., unit vetor i ˆ ) is related to unit vetors in the x y oordinate system as follows: iˆ = iˆ + ˆj + kˆ (.7) x x x y x As i ˆ is a unit vetor, then in aordane with Equation.: ( ) + ( ) + ( ) = 1 (.8) x x x y x To this point we have referred to a fore as a vetor. A fore vetor an also be alled a fore tensor. The term tensor refers to any quantity that transforms in a physially meaningful way from one Cartesian oordinate system to another. The rank of a tensor equals the number of subsripts that must be used to desribe the tensor. A fore an be desribed using a single subsript, F i, and therefore a fore is said to be a tensor of rank one, or equivalently, a firstorder tensor. Equation.6 is alled the transformation law for a first-order tensor. It is likely that the reader is already familiar with two other tensors: the stress tensor, σ ij, and the strain tensor, ε ij. The stress and strain tensors will be reviewed later in this hapter, but at this point it an be noted that two subsripts are used to desribe stress and strain tensors. Hene, stress and strain tensors are said to be tensors of rank two, or equivalently, seond-order tensors. Example Problem.1 Given: All omponents of a fore vetor F are known in a given x y oordinate system. It is desired to express this fore in a new x y oordinate system, where the x y system is generated from the original x y system by the following two rotations (see Figure.): A rotation of θ-degrees about the original -axis (whih defines an intermediate x y oordinate system), followed by A rotation of β-degrees about the x -axis (whih defines the final x y oordinate system) Problem a. Determine the diretion osines i j relating the original x y oordinate system to the new x y oordinate system. b. Obtain a general expression for the fore vetor F in the x y oordinate system. K13483_C00.indd 44 9/13/01 11:04:18 AM

5 Review of Fore, Stress, and Strain Tensors 45 (a) +x +θ +x +y +y +θ +, + +θ (b) +x, +x +β +y +y +β + +β + Figure. Generation of the x y oordinate system from the x y oordinate system. (a) Rotation of θ-degrees about the original -axis (whih defines an intermediate x y oordinate system); (b) rotation of β-degrees about the x -axis (whih defines the final x y oordinate system).. Calulate numerial values of the fore vetor F in the x y oordinate system if θ = 0, β = 60, and F x = 1000 N, F y = 00 N, F = 600 N. Solution a. One way to determine diretion osines i j is to rotate unit vetors. In this approah unit vetors are first rotated from the original x y oordinate system to the intermediate x y oordinate system, and then from the x y system to the final x y oordinate system. Define a unit vetor I that is aligned with the x-axis: I ( 1) i That is, vetor I is a vetor for whih I x = 1, I y = 0, and I = 0. The vetor I an be rotated to the intermediate x y oordinate system using Equation.6: K13483_C00.indd 45 9/13/01 11:04:19 AM

6 46 Strutural Analysis of Polymeri Composite Materials I = I + I + I x x x x x y y x I = I + I + I y y x x y y y y I = I + I + I x x y y The diretion osines assoiated with a transformation from the x y oordinate system to the intermediate x y oordinate system an be determined by inspetion (see Figure.a), and are given by = osine(angle between x - and x-axes) = os θ x x x y x = osine( angle between x - and y-axes) = os( 90 θ) = sin θ = osine( angle between x - and -axes) = os( 90 ) = 0 y x y y = osine(angle between the y - and x-axes) = os( 90 + θ) = sin θ = osine( angle between the y - and y-axes) = os θ = osine(angle between the y - and -axes) = os( 90 ) = 0 y x y = osine(angle between the - and x-axes) = os( 90 ) = 0 = osine(angle between the - and y-axes) = os( 90 ) = 0 = osine(angle between the - and -axes) = os( 0 ) = 1 Using these diretion osines: Ix = x xix + x yiy + x I = (os θ)( 1) + (sin θ)( 0) + ( 0)( 0) = os θ Iy = y xix + y yiy + y I = ( sin θ)( 1) + (os θ)( 0) + ( 0)( 0) = sinθ I = xix + yiy + I = ( 0)( 1) + ( 0)( 0) + ( 1)( 0) = 0 Therefore, in the x y oordinate system, the vetor I is written: I = (os θ) i + ( sin θ) j Now define two additional unit vetors, one aligned with the original y-axis (vetor J ) and one aligned with the original -axis (vetor K ); that is, let J = ( 1 ) j and K = ( 1 ) k. Transforming these vetors to the x y oordinate system, again using the diretion osines listed above, results in J = (sin θ) i + ( os θ) j K13483_C00.indd 46 9/13/01 11:04:19 AM

7 Review of Fore, Stress, and Strain Tensors 47 K = ( 1 ) k We now rotate vetors I, J and K from the intermediate x y oordinate system to the final x y oordinate system. The diretion osines assoiated with a transformation from the x y oordinate system to the final x y oordinate system are easily determined by inspetion (see Figure.b) and are given by = 1 = 0 = 0 x x x y x = 0 = os β = sin β y x y y y = 0 = sin β = os β x y These diretion osines together with Equation.6 an be used to rotate the vetor I from the intermediate x y oordinate system to the final x y oordinate system: I = I + I + I = ( 1)(os θ) + ( 0)( sin θ) + ( 0)( 0) x x x x x y y x I x = os θ I = I + I + I = ( 0)(os θ) + (os β)( sin θ) + (sin β)( 0) y y x x y y y y I y = os β sin θ I = I + I + I = ( 0)(os θ) + ( sin β)( sin θ) + (os β)( 0) x x y y I = sin β sin θ Therefore, in the final x y oordinate system, the vetor I is written: I = (os θ) i + ( os βsin θ) j + (sin β sin θ) k (a) Reall that in the original x y oordinate system I is simply a unit vetor aligned with the original x-axis: I ( 1 ) i. Therefore, result (a) defines the diretion osines assoiated with the angle between the original x-axis and the final x -, y -, and -axes. That is x x = osθ y x x = os β sin θ = sin β sin θ K13483_C00.indd 47 9/13/01 11:04:19 AM

8 48 Strutural Analysis of Polymeri Composite Materials A similar proedure is used to rotate the unit vetors J and K from the intermediate x y oordinate system to the final x y oordinate system. These rotations result in J = (sin θ) i + (os βos θ) j + ( sin β os θ) k (b) K = ( 0) i + (sin β) j + (os β ) k () As vetor J is a unit vetor aligned with the original y-axis, J = ( 1) j, result (b) defines the diretion osines assoiated with the angle between the original y-axis and the final x -, y -, and -axes: = sin θ x y y y y = os β os θ = sin β os θ Finally, result () defines the diretion osines assoiated with the angle between the original -axis and the final x -, y -, and -axes: = 0 x y = sin β = os β Assembling the preeding results, the set of diretion osines relating the original x y oordinate system to the final x y oordinate system an be written: x x x y x y x y y y x y osθ sinθ 0 = osβ sinθ osβ osθ sinβ sinβ sinθ sinβ osθ osβ b. As diretion osines have been determined, transformation of fore vetor F an be aomplished using any version of Equation.6. For example, using matrix notation, Equation.6: Fx Fy = F x x x y x y x y y y x y Fx Fy F os θ sin θ 0 Fx = os β sin θ os β os θ sin β Fy sin β sin θ sin β os θ os β F K13483_C00.indd 48 9/13/01 11:04:0 AM

9 Review of Fore, Stress, and Strain Tensors 49 Fx (os θ) Fx + (sin θ) Fy Fy = ( osβsin θ) Fx + (osβos θ) Fy + (sin β) F F (sinβsin θ) Fx + ( sinβos θ) Fy + (os β) F. Using the speified numerial values and the results of part (b): [os( 0 )]( 1000 N) + [sin( 0 )] 00 N Fx [ os( 60 )(sin 0 )]( 1000 N) + [os( 60 )os( 0 )]( 00 N) Fy = + [sin( 60 )]( 600 N) F [sin( 60 )sin( 0 )]( 1000 N) + [ sin( 60 )os( 0 )]( 00 N) + (os 60 )( 600 N) Fx N Fy = N F N Using vetor notation, F an now be expressed in the two different oordinate systems as or, equivalently F = (1000 N) iˆ + (00 N) ˆj + ( 600 N) kˆ F = (873.1 N) i + (784.6 N) j + ( N) k Where iˆ, ˆ, j k ˆ and i ˆ, ˆ j, k ˆ are unit vetors in the x y and x y oordinate systems, respetively. Fore vetor F drawn in the x y and x y oordinate systems is shown in Figure.3a and b, respetively. The two desriptions of F are entirely equivalent. A onvenient way of (partially) verifying this equivalene is to alulate the magnitude of the original and transformed fore vetors. As the magnitude is an invariant, it is independent of the oordinate system used to desribe the fore vetor. Using Equation., the magnitude of the fore vetor in the x y oordinate system is F = F + F + F = ( 1000 N) + ( 00 N) + ( 600 N) = 1183 N x y The magnitude of the fore vetor in the x y oordinate system is x y F = ( F ) + ( F ) + ( F ) = ( N) + ( N) + ( N) = 1183 N (agrees) K13483_C00.indd 49 9/13/01 11:04:0 AM

10 50 Strutural Analysis of Polymeri Composite Materials (a) +x +y F F x = 1000 N F = 600 N F y = 00 N + (b) F +x F x = N +y F y = N F = N + Figure.3 Fore vetor F drawn in two different oordinate systems. (a) Fore vetor F in the original x y oordinate system; (b) fore vetor F in a new x y oordinate system..3 Normal Fores, Shear Fores, and Free-Body Diagrams Fore F ating at an angle to a planar surfae is shown in Figure.4. As fore is a vetor, it an always be deomposed into two fore omponents, a normal fore omponent and a shear fore omponent. The line-of-ation of the normal fore omponent is orthogonal to the surfae, whereas the line-of-ation of the shear fore omponent is tangent to the surfae. Internal fores indued within a solid body by externally applied fores an be investigated with the aid of free-body diagrams. A simple example is shown in Figure.5, whih shows a straight irular rod with onstant diameter subjeted to two external fores of equal magnitude (R) but opposite diretion. The internal fore (FI, say) indued at any ross-setion of the rod an be investigated by making an imaginary ut along the plane of interest. Suppose an imaginary ut is made along plane a a a a, whih is perpendiular to the axis of the rod. The resulting free-body diagram for the lower half of the rod is shown in Figure.5a, where a x y oordinate system has been assigned suh that the x-axis is parallel to the rod axis, as shown. On the basis of this free-body diagram, it is onluded that an internal fore FI = ( R) i + ( 0) j + ( 0) k is indued at ross-setion a a. That K13483_C00.indd 50 9/13/01 11:04:0 AM

11 Review of Fore, Stress, and Strain Tensors 51 F Normal fore omponent Shear fore omponent Planar surfae Figure.4 A fore F ating at an angle to a planar surfae. (a) R +x R a a a a +y + R R (b) R +x +θ +x a b b a a b b a +θ R os θ +y +θ +y +,+ R os θ R R Figure.5 The use of free-body diagrams to determine internal fores ating on planes a a a a and b b b b. (a) Free-body diagram based on plane a a a a, perpendiular to rod axis; (b) free-body diagram based on plane b b b b, inlined at angle +θ to the rod axis. is, only a normal fore of magnitude R is indued at ross-setion a a a a, whih has been defined to be perpendiular to the axis of the rod. In ontrast, the imaginary ut need not be made perpendiular to the axis of the rod. Suppose the imaginary ut is made along plane b b b b, whih is inlined at an angle of +θ with respet to the axis of the rod. The resulting K13483_C00.indd 51 9/13/01 11:04:0 AM

12 5 Strutural Analysis of Polymeri Composite Materials free-body diagram for the lower half of the rod is shown in Figure.5b. A new x y oordinate system has been assigned so that the x -axis is perpendiular to plane b b b b and the -axis is oinident with the -axis, that is, the x y oordinate system is generated from the x y oordinate system by a rotation of +θ about the original -axis. The internal fore FI an be expressed with respet to the x y oordinate system by transforming FI from the x y oordinate system to the x y oordinate system. This oordinate transformation is a speial ase of the transformation onsidered in Example Problem.1. The diretion osines now beome (with β = 0 ): = os θ = sin θ = 0 = sin θ = os θ = 0 x x x y x y x y y y = 0 x y Applying Equation.6, we have Fx Fy = F x x x y x y x y y y x y = 0 = 1 Fx os θ sin θ 0 R (os θ) R Fy = sin θ os θ 0 0 = ( sin θ) R F In the x y oordinate system the internal fore is FI = ( Ros θ) i ( Rsin θ ) j + ( 0 ) k. Hene, by defining a oordinate system whih is inlined to the axis of the rod, we onlude that both a normal fore (R os θ) and a shear fore ( R sin θ) are indued in the rod. Although the preeding disussion may seem simplisti, it has been inluded to demonstrate the following: A speifi oordinate system must be speified before a fore vetor an be defined in a mathematial sense. In general, the oordinate system is defined by the imaginary ut(s) used to form the free-body diagram. All omponents of a fore must be speified to fully define the fore vetor. Further, the individual omponents of a fore hange as the vetor is transformed from one oordinate system to another. These two observations are valid for all tensors, not just for fore vetors. In partiular, these observations hold in the ase of stress and strain tensors, whih will be reviewed in the following setions..4 Definition of Stress There are two fundamental types of stress: normal stress and shear stress. Both types of stress are defined as a fore divided by the area over whih it ats. K13483_C00.indd 5 9/13/01 11:04:1 AM

13 Review of Fore, Stress, and Strain Tensors 53 A general 3D solid body subjeted to a system of external fores is shown in Figure.6a. It is assumed that the body is in stati equilibrium, that is, it is assumed that the sum of all external fores is ero, ΣFi = 0. These external fores indue internal fores ating within the body. In general, the internal fores will vary in both magnitude and diretion throughout the body. An illustration of the variation of internal fores along a single line within an internal plane is shown in Figure.6b. A small area (ΔA) isolated from this plane is shown in Figure.6. Area ΔA is assumed to be infinitesimally small. That (a) F 1 F F 3 F 5 F 4 (b) Internal fores along a single line F 3 F 5 F 4 () V N ΔA F 3 F 5 F 4 Figure.6 A solid 3-D body in equilibrium. (a) A solid 3-D body subjet to external fores F1 F5; (b) variation of internal fores along an internal line; () internal fore ating over infinitesimal area ΔA. K13483_C00.indd 53 9/13/01 11:04:3 AM

14 54 Strutural Analysis of Polymeri Composite Materials is, the area ΔA is small enough suh that the internal fores ating over ΔA an be assumed to be of onstant magnitude and diretion. Therefore, the internal fores ating over ΔA an be represented by a fore vetor whih an be deomposed into a normal fore, N, and a shear fore, V, as shown in Figure.6. Normal stress (usually denoted σ ), and shear stress (usually denoted τ ) are defined as the fore per unit area ating perpendiular and tangent to the area ΔA, respetively. That is, N σ lim A V τ lim A A 0 A 0 (.9) Note that by definition the area ΔA shrinks to ero: ΔA 0. Stresses σ and τ are therefore said to exist at a point. As internal fores generally vary from point-to-point (as shown in Figure.6), stresses also vary from point-to-point. Stress has units of fore per unit area. In SI units stress is reported in terms of Pasals (abbreviated Pa ), where 1 Pa = 1 N/m. In English units stress is reported in terms of pounds-fore per square inh (abbreviated psi ), that is, 1 psi = 1 lbf/in. Conversion fators between the two systems of measurement are 1 psi = 6895 Pa, or equivalently, 1 Pa = psi. Common abbreviations used throughout this hapter are as follows: Pa = 1 kilo-pasals = 1 kpa psi = 1 kilo-psi = 1 ksi Pa = 1 Mega-Pasals = 1 MPa psi = 1 mega-psi = 1 Msi Pa = 1 Giga-Pasals = 1 GPa..5 The Stress Tensor A general 3D solid body subjeted to a system of external fores is shown in Figure.7a. It is assumed that the body is in stati equilibrium and that body fores are negligible, that is, is it assumed that the sum of all external fores is ero, ΣFi = 0. A free-body diagram of an infinitesimally small ube removed from the body is shown in Figure.7b. The ube is referened to a x y oordinate system, and the ube edges are aligned with these axes. The lengths of the ube edges are denoted dx, dy, and d. Although (in general) internal fores are indued over all six faes of the ube, for larity the fores ating on only three faes have been shown. The fore ating over eah ube fae an be deomposed into a normal fore omponent and two shear fore omponents, as illustrated in Figure.7. Although eah fore omponent ould be identified with a single subsript (as fore is a first-order tensor), for onveniene two subsripts have been used. The first subsript identifies the fae over whih the fore is distributed, whereas the seond subsript identifies the diretion in whih the K13483_C00.indd 54 9/13/01 11:04:3 AM

15 Review of Fore, Stress, and Strain Tensors 55 (a) (b) F 1 +x F +y F 3 F 5 F 4 + () V xy N xx +x (d) τ xy σ xx +x V x V +y yx +y τ x τ yx V y τ y V y τ y N N yy σ σ yy + V x + τ x Figure.7 Free-body diagrams used to define stress indued in a solid body. (a) 3-D solid body in equilibrium; (b) infinitesimal ube removed from the solid body (internal fores ating on three faes shown); () normal fore and two shear fores at over eah fae of the ube; (d) normal stress and two shear stresses at over eah fae of the ube. fore is oriented. For example, N xx refers to a normal fore omponent whih is distributed over the x-fae and whih is ating parallel to the x-diretion. Similarly, V y refers to a shear fore distributed over the -fae whih is ating parallel to the y-diretion. Three stress omponents an now be defined for eah ube fae, in aordane with Equation.9. For example, for the three faes of the infinitesimal element shown in Figure.7: Stresses ating on the -x-fae: Nxx Vxy σxx = lim τxy τ y y = lim d, d 0 d d dy, d 0 dy d x V = lim x dy, d 0 dy d Stresses ating on the y-fae: Nyy Vyx σ yy = lim τyx τ x x = lim d, d 0 d d dx, d 0 dx d y V y = lim d x, d 0 dx d K13483_C00.indd 55 9/13/01 11:04:4 AM

16 56 Strutural Analysis of Polymeri Composite Materials Stresses ating on the +-fae: N Vx V y σ = lim τx τy x y x y = lim x y x y = lim d, d 0 d d d, d 0 d d d x, d y 0 dx dy As three fore omponents (and therefore three stress omponents) exist on eah of the six faes of the ube, it would initially appear that there are 18 independent fore (stress) omponents. However, it is easily shown that for stati equilibrium to be maintained (assuming body fores are negligible): Normal fores ating on opposite faes of the infinitesimal element must be of equal magnitude and opposite diretion. Shear fores ating within a plane of the element must be orientated either tip-to-tip (e.g., fores V x and V x in Figure.7) or tail-totail (e.g., fores V xy and V yx ), and be of equal magnitude. That is V xy = V yx, V x = V x, V y = V y. These restritions redue the number of independent fore (stress) omponents from 18 to 6, as follows: Independent Fore Component(s) Corresponding Stress Component(s) N xx N yy σ xx σ yy N σ V xy (=V yx ) τ xy (= τ yx ) V x (=V x ) τ x (= τ x ) V y (=V y ) τ y (= τ y ) We must next define the algebrai sign onvention we will use to desribe individual stress omponents. The omponents ating on three faes of an infinitesimal element are shown in Figure.8. We first assoiate an algebrai sign with eah fae of the infinitesimal element. A ube fae is positive if the outward unit normal of the fae (i.e., the unit normal pointing away from the interior of the element) points in a positive oordinate diretion; otherwise, the fae is negative. For example, faes (ABCD) and (ADEG) in Figure.8 are a positive faes, whereas fae (DCFE) is a negative fae. Having identified the positive and negative faes of the element, a stress omponent is positive if The stress omponent ats on a positive fae and points in a positive oordinate diretion, or if The stress omponent ats on a negative fae and points in a negative oordinate diretion. If neither of these onditions are met, then the stress omponent is negative. This onvention an be used to onfirm that all stress omponents K13483_C00.indd 56 9/13/01 11:04:4 AM

17 Review of Fore, Stress, and Strain Tensors 57 +x σ xx B +y A τ x C τ y τ xy D τ yx τ y σ G τ x F σ yy E + Figure.8 An infinitesimal stress element (all stress omponents shown in a positive sense). shown in Figure.8 are algebraially positive. For example, to determine the algebrai sign of the normal stress σ xx whih ats on fae ABCD in Figure.8, note that (a) fae ABCD is positive, and (b) the normal stress σ xx whih ats on this fae points in the positive x-diretion. Therefore, σ xx is positive. As a seond example, the shear stress τ y whih ats on ube fae DCFE is positive beause (a) fae DCFE is a negative fae, and (b) τ y points in the negative -diretion. The preeding disussion shows that the state of stress at a point is defined by six omponents of stress: three normal stress omponents and three shear stress omponents. The state of stress is written using matrix notation as follows: σ τ τ τ σ τ τ τ σ xx xy x yx yy y x y σ τ τ = τ σ τ τx τ σ xx xy x xy yy y y (.10) To express the state of stress using indiial notation we must first make the following hange in notation: τ σ xy xy τ τ τ τ τ x yx y x y σ σ σ σ σ x yx y x y K13483_C00.indd 57 9/13/01 11:04:5 AM

18 58 Strutural Analysis of Polymeri Composite Materials With this hange the matrix on the left side of the equality sign in Equation.10 beomes: σ τ τ τ σ τ τ τ σ xx xy x yx yy y x y σ σ σ σ σ σ σ σ σ xx xy x yx yy y x y Whih an be suintly written using indiial notation as σ ij, i, j = x, y, or (.11) In Setion.1 it was noted that a fore vetor is a first-order tensor, as only one subsript is required to desribe a fore tensor, F i. From Equation.11 is it lear that stress is a seond-order tensor (or equivalently, a tensor of rank two), as two subsripts are required to desribe a state of stress. Example Problem. Given: The stress element referened to a x y oordinate system and subjet to the stress omponents shown in Figure.9. Determine: Label all stress omponents, inluding algebrai sign. Solution The magnitude and algebrai sign of eah stress omponent is determined using the sign onvention defined above. The proedure will +x B 100 +y A D C 5 G 30 F 50 E + Figure.9 Stress omponents ating on an infinitesimal element (all stresses in MPa). K13483_C00.indd 58 9/13/01 11:04:6 AM

19 Review of Fore, Stress, and Strain Tensors 59 be illustrated using the stress omponents ating on fae DCFE. First, note that fae DCFE is a negative fae, as an outward unit normal for this fae points in the negative y-diretion. The normal stress whih ats on fae DCFE has a magnitude of 50 MPa, and points in the positive y-diretion. Hene, this stress omponent is negative and is labeled σ yy = 50 MPa. One of the shear stress omponents ating on fae DCFE has a magnitude of 75 MPa, and points in the positive x-diretion. Hene, this stress omponent is also negative and is labeled τ yx = 75 MPa (or equivalently, τ xy = 75 MPa). Finally, the seond shear fore omponent ating on fae DCFE has a magnitude of 50 MPa, and points in the positive -diretion. Hene, this omponent is labeled τ y = 50 MPa (or equivalently, τ y = 50 MPa). Following this proess for all faes of the element, the state of stress represented by the element shown in Figure.9 an be written: σ τ τ τ σ τ τ τ σ xx xy x yx yy y x y 100 MPa 75MPa 30 MPa = 75MPa 50 MPa 50 MPa 30 MPa 50 MPa 5 MPa.6 Transformation of the Stress Tensor In Setion.5 the stress tensor was defined using a free-body diagram of an infinitesimal element removed from a 3D body in stati equilibrium. This onept is again illustrated in Figure.10a, whih shows the stress element referened to an x y oordinate system. Now, the infinitesimal element need not be removed in the orientation shown in Figure.10a. An infinitesimal element removed from preisely the same point within the body but at a different orientation is shown in Figure.10b. This stress element is referened to a new x y oordinate system. The state of stress at the point of interest is ditated by the external loads applied to the body, and is independent of the oordinate system used to desribe it. Hene, the stress tensor referened to the x y oordinate system is equivalent to the stress tensor referened to the x y oordinate system, although the diretion and magnitude of individual stress omponents will differ. The proess of relating stress omponents in one oordinate system to those in another is alled transformation of the stress tensor. This terminology is perhaps unfortunate, in the sense that the state of stress itself is not transformed but rather our desription of the state of stress transforms as we hange from one oordinate system to another. K13483_C00.indd 59 9/13/01 11:04:6 AM

20 60 Strutural Analysis of Polymeri Composite Materials (a) F 1 F +x τ xy σ xx τ x +y τ yx τ y F 3 τ y σ σ yy F 5 F 4 + τ x (b) F 1 σ τ F x x x +x +y τ xy τ y x F 3 τ y τ y σ y y σ τ x F 5 + F 4 Figure.10 Infinitesimal elements removed from the same point within a 3D solid but in two different orientations. (a) Infinitesimal element referened to the x y oordinate system; (b) infinitesimal element referened to the x y oordinate system. It an be shown [1,] that the stress omponents in the new x y oordinate system (σ i j ) are related to the omponents in the original x y oordinate system (σ ij ) aording to σi j = i k j l σkl where i, j, k, l = x, y, (.1a) Or, equivalently (using matrix notation): [ σ ] = [ ][ σ ][ ] T i j i j ij i j K13483_C00.indd 60 9/13/01 11:04:7 AM

21 Review of Fore, Stress, and Strain Tensors 61 where [ i j ] T is the transpose of the diretion osine array. Writing in full matrix form: σ σ σ σ σ σ σ σ σ x x x y x y x y y y x y = x x x y x y x y y y x y σ σ σ σ yx σ yy σ σ σ σ xx xy x y x y x x y x x x y y y y x y (.1b) As disussed in Setion., the terms i j whih appear in Equation.1a,b are diretion osines and equal the osine of the angle between the axes of the x y and x y oordinate systems. Reall that the algebrai sign of an angle of rotation is defined in aordane with the right-hand rule, and that angles are defined from the x y oordinate system to x y oordinate system. Equation.1a,b is alled the transformation law for a seondorder tensor. If an analysis is being performed with the aid of a digital omputer, whih nowadays is almost always the ase, then matrix notation (Equation.1b) will most likely be used to transform a stress tensor from one oordinate system to another. Conversely, if a stress transformation is to be aomplished using hand alulations, then indiial notation (Equation.1a) may be the preferred hoie. To apply Equation.1a, the stress omponent of interest is speified by seleting the appropriate values for subsripts i and j, and then the terms on the right side of the equality are summed over the entire range of the remaining two subsripts, k and l. For example, suppose we wish write the relationship between σ x and the stress omponents in the x y oordinate system in expanded form. We first speify that i = x and j =, and Equation.1a beomes: σx = x k l σkl where k, l = x, y, We then sum all terms on the right side of the equality formed by yling through the entire range of k and l. In expanded form, we have σ = σ + σ + σ x x x x xx x x y xy x x x + σ + σ + σ + x y x yx x y y yy x y y σ + σ + σ (.13) x x x x y y x K13483_C00.indd 61 9/13/01 11:04:8 AM

22 6 Strutural Analysis of Polymeri Composite Materials Equations.1a and.1b show that the value of any individual stress omponent σ i j varies as the stress tensor is transformed from one oordinate system to another. However, it an be shown [1,] that there are features of the total stress tensor that do not vary when the tensor is transformed from one oordinate system to another. These features are alled the stress invariants. For a seond-order tensor three independent stress invariants exist, and are defined as follows: First stress invariant = Θ = σ ii (.14a) Seond stress invariant = Φ = 1 ( σ σ σ σ ) ii jj ij ij (.14b) Third stress invariant = Ψ = 1 (σ σ σ 3σ σ σ + σ σ σ 6 ii jj kk ii jk jk ij jk ki) (.14) Alternatively, by expanding these equations over the range i,j,k = x,y, and simplifying, the stress invariants an be written: First stress invariant = Θ = σxx + σ yy + σ (.15a) Seond stress invariant = Φ = σ σ + σ σ + σ σ ( σ + σ + xx yy xx yy xy x Thirdstressinvariant = Ψ = σ σ σ σ σ σ σ σ σ + xx yy xx y yy x xy σ y ) (.15b) σ σ σ xy x y (.15) The three stress invariants are oneptually similar to the magnitude of a fore tensor. That is, the value of the three stress invariants is independent of the oordinate used to desribe the stress tensor, just as the magnitude of a fore vetor is independent of the oordinate system used to desribe the fore. This invariane will be illustrated in the following Example Problem. Example Problem.3 Given: A state of stress referened to a x y oordinate is known to be σ σ σ σ σ σ σ σ σ xx xy x yx yy y x y = ( ksi) K13483_C00.indd 6 9/13/01 11:04:9 AM

23 Review of Fore, Stress, and Strain Tensors 63 It is desired to express this state of stress in an x y oordinate system, generated by the following two sequential rotations: i. Rotation of θ = 0 about the original -axis (whih defines an intermediate x y oordinate system), followed by ii. Rotation of β = 35 about the x -axis (whih defines the final x y oordinate system) Problem a. Rotate the stress tensor to the x y oordinate system. b. Calulate the first, seond, and third invariants of the stress tensor using (i) elements of the stress tensor referened to the x y oordinate system, σ ij, and (ii) elements of the stress tensor referened to the x y oordinate system, σ i j. Solution a. General expressions for diretion osines relating the x y and x y oordinate systems were determined as a part of Example Problem.1. The diretion osines were found to be x x x y x y x y y y x y = osθ = sin θ = 0 = os β sin θ = os β os θ = sin β = sin β sin θ = sin β os θ = os β As in this problem θ = 0 and β = 35, the numerial values of the diretion osines are x x x y x y x = os( 0 ) = = sin( 0 ) = = 0 = os( 35 )sin( 0 ) = y y = os( 35 )os( 0 ) = y = sin( 35 ) = x = sin( 35 )sin( 0 ) = y = sin( 35 )os( 0 ) = = os( 35 ) = K13483_C00.indd 63 9/13/01 11:04:30 AM

24 64 Strutural Analysis of Polymeri Composite Materials Eah omponent of the transformed stress tensor is now found through appliation of either Equation.1a or.1b. For example, if indiial notation is used stress omponent σ x an be found using Equation.13: σ = σ + σ + σ x x x x xx x x y xy x x x + σ + σ + σ x y x yx x y y yy x y y + σ + σ + σ x x x x y y x σ x = ( )( ) ( 50ksi) + ( )( ) ( 10ksi) + ( )( ) ( 15ksi) + ( ) ( ) ( 10ksi) + ( )( ) ( 5ksi) + ( )( ) ( 30ksi) + ( 0) ( ) ( 15ksi) + ( 0)( )( 30ksi) + ( 0)( )( 5ksi) σ x = 8. 95ksi Alternatively, if matrix notation is used, then Equation.1b beomes: σ σ σ σ σ σ σ σ σ x x x y x y x y y y x y = Completing the matrix multipliation indiated, there results: σ σ σ σ σ σ σ σ σ x x x y x y x y y y x y = ( ksi) Notie that the value of σ x determined through matrix multipliation is idential to that obtained using indiial notation, as previously desribed. The stress element is shown in the original and final oordinate systems in Figure.11. b. The first, seond, and third stress invariants will now be alulated using omponents of both σ ij and σ i j. It is expeted K13483_C00.indd 64 9/13/01 11:04:31 AM

25 Review of Fore, Stress, and Strain Tensors 65 (a) 50 +x (b) x y y Figure.11 Stress tensor of Example Problem.3, referened to two different oordinate systems (magnitude of all stress omponents in ksi). (a) Referened to x y oordinate system; (b) referened to x y oordinate system. that idential values will be obtained, as the stress invariants are independent of oordinate system. First Stress Invariant: x y oordinate system: Θ = σ = σ + σ + σ ii xx yy Θ = ( ) ksi Θ = 70 ksi x y oordinate system: Θ = σ = σ + σ + σ i i x x y y Θ = ( ) Θ = 70ksi As expeted, the first stress invariant is independent of oordinate system. Seond Stress Invariant: x y oordinate system: Φ = ( ) 1 ( σ σ σ σ ) = σ σ + σ σ + σ σ σ + σ + σ ii jj ij ij xx yy xx yy xy x y K13483_C00.indd 65 9/13/01 11:04:3 AM

26 66 Strutural Analysis of Polymeri Composite Materials { + + } Φ = ( 50)( 5) + ( 50)( 5) + ( 5)( 5) [( 10) ( 15) ( 30) ] ( ksi) Φ = 350( ksi) x y oordinate system: Φ = 1 ( ) σ i i σ j j σ i j σ i j ( ) Φ = σ σ + σ σ + σ σ σ + σ + σ x x y y x x y y x y x y { Φ = ( )( ) + ( )( 13. 7) + ( )( 13. 7) [( ) + ( 8. 95) + ( ) ]}( ksi) Φ = 350( ksi) As expeted, the seond stress invariant is independent of oordinate system. Third Stress Invariant: x y oordinate system: 1 Ψ = σ σ σ σ σ σ σ σ σ ( ii jj kk ii jk jk ij jk ki ) Ψ = σ σ σ σ σ σ σ σ σ + σ σ σ xx yy xx y yy x xy xy x y Ψ = [( 50)( 5)( 5) ( 50)( 30) ( 5)( 15) ( 5)( 10) + ( 10)( 15)( 30)]( ksi) 3 Ψ = 65375( ksi) 3 x y oordinate system: 1 Ψ = σ σ σ σ σ σ σ σ σ ( i i j j k k i i j k j k i j j k k i ) Ψ = σ σ σ σ σ σ σ σ σ + σ x x y y x x y y y x x y x y σx σ y K13483_C00.indd 66 9/13/01 11:04:36 AM

27 Review of Fore, Stress, and Strain Tensors 67 Ψ = [( )( )( 13. 7) ( )( ) ( )( 8. 95) ( 13. 7)( ) + ( )( 8. 95)( )]( ksi) 3 Ψ = 65375( ksi) 3 As expeted, the third stress invariant is independent of oordinate system..7 Prinipal Stresses It is always possible to rotate the stress tensor to a speial oordinate system in whih no shear stresses exist. This oordinate system is alled the prinipal stress oordinate system, and the normal stresses that exist in this oordinate system are alled prinipal stresses. Prinipal stresses an be used to predit failure of isotropi materials. Therefore, knowledge of the prinipal stresses and orientation of the prinipal stress oordinate system is of vital importane during design and analysis of isotropi metal strutures. This is not the ase for anisotropi omposite materials. Failure of omposite material is not governed by prinipal stresses. Prinipal stresses are only of oasional interest to the omposite engineer and are reviewed here only in the interests of ompleteness. Prinipal stresses are usually denoted as σ 1, σ, and σ 3. However, in the study of omposites, the labels 1,, and 3 are used to label a speial oordinate system alled the prinipal material oordinate system. Therefore, in this hapter the axes assoiated with the prinipal stress oordinate system will be labeled the p 1, p, and p 3 axes, and the prinipal stresses will be denoted as σ p1, σ p, and σ p3. Prinipal stresses may be related to stress omponents in an x y oordinate system using the free-body diagram shown in Figure.1. It is assumed that plane ABC is one of the three prinipal planes (i.e., n = 1,, or 3), and therefore no shear stress exists on this plane. The line-of-ation of prinipal stress σ pn defines one axis of the prinipal stress oordinate system. The diretion osines between this prinipal axis and the x-, y-, and -axes are pnx, pny, and pn, respetively. The surfae area of triangle ABC is denoted A ABC. The normal fore ating over triangle ABC therefore equals (σ pn )(A ABC ). The omponents of this normal fore ating in the x-, y-, and -diretions equal ( pnx )(σ pn )(A ABC ), ( pny )(σ pn )(A ABC ) and ( pn )(σ pn )(A ABC ), respetively. K13483_C00.indd 67 9/13/01 11:04:36 AM

28 68 Strutural Analysis of Polymeri Composite Materials +x σ pn τ y σ τ x C +y B τ xy τ yx D σ yy τ y τ x + A σ xx Figure.1 Free-body diagram used to relate strress omponents in the x y oordinate system to a prinipal stress. The area of the other triangular faes are given by Area of triangle ABD = ( pnx )(A ABC ) Area of triangle ACD = ( pny )(A ABC ) Area of triangle BCD = ( pn )(A ABC ). Summing fores in the x-diretion and equating to ero, we obtain σ A σ A τ A τ A = 0 pnx pn ABC xx pnx ABC xy pny ABC x pn ABC whih an be redued and simplified to ( σ pn σxx) pnx τxypny τxpn = 0 (.16a) Similarly, summing fores in the y- and -diretions results in τxypnx + ( σ pn σ yy) pny τypn = 0 (.16b) τxpnx τypny + ( σ pn σ) pn = 0 (.16) Equation.16 represent three linear homogeneous equations whih must be satisfied simultaneously. As diretion osines pnx, pny, and pn must also satisfy Equation.8, and therefore annot all equal ero, the solution an be obtained by requiring that the determinant of the oeffiients of pnx, pny, and pn equal ero: ( σ σ ) τ τ pn xx xy x τ ( σ σ ) τ xy pn yy y τ τ ( σ σ ) x y pn = 0 K13483_C00.indd 68 9/13/01 11:04:38 AM

29 Review of Fore, Stress, and Strain Tensors 69 Equating the determinant to ero results in the following ubi equation: 3 σ Θσ + Φσ Ψ = 0 (.17) pn pn pn where Θ, Φ, and Ψ are the first, seond, and third stress invariants, respetively, and have been previously listed as Equations.14 and.15. The three roots of this ubi equation represent the three prinipal stresses and may be found by appliation of the standard approah [3]. By onvention, the prinipal stresses are numbered suh that σ p1 is the algebraially greatest prinipal stress, whereas σ p3 is the algebraially least. That is, σ p1 > σ p > σ p3. One the prinipal stresses are determined the three sets of diretion osines (whih define the prinipal oordinate diretions) are found by substituting the three prinipal stresses given by Equation.17 into Equation.16 in turn. As only two of Equation.16 are independent, Equation.8 is used as a third independent equation involving the three unknown onstants, pnx, pny }, and pn. The proess of finding prinipal stresses and diretion osines will be demonstrated in the following Example Problem. Example Problem.4 Given: A state of stress referened to an x y oordinate is known to be: σ σ σ σ σ σ σ σ σ xx xy x yx yy y x y = ( ksi) Problem Find (a) the prinipal stresses and (b) the diretion osines that define the prinipal stress oordinate system. Solution This is the same stress tensor onsidered in Example Problem.3. As a part of that problem the first, seond, and third stress invariants were found to be Θ = 70 ksi Φ = 350 ( ksi) Ψ = ( ksi) 3 K13483_C00.indd 69 9/13/01 11:04:40 AM

30 70 Strutural Analysis of Polymeri Composite Materials a. Determining the Prinipal Stresses: In aordane with Equation.17, the three prinipal stresses are the roots of the following ubi equation: 3 σ 70σ 350σ = 0 The three roots of this equation represent the three prinipal stresses, and are given by σ = ksi, σ = ksi, and σ = 7. 7 ksi p1 p p3 b. Determining the Diretion Cosines: The first two of Equations.16 and.8 are used to form three independent equations in three unknowns. We have ( σ σ ) τ τ = 0 pn xx pnx xy pny x pn τ + ( σ σ ) τ = 0 xy pnx pn yy pny y pn ( ) + ( ) + ( ) = 1 pnx pny pn Diretion osines for σ p1 : The three independent equations beome: ( ) = 0 p1x p1y p ( ) 30 = 0 p1x p1y p1 ( ) + ( ) + ( ) = 1 p1x p1y p1 Solving simultaneously, we obtain: p1x = p1y = p1 = Diretion osines for σ p : The three independent equations beome: ( ) = 0 px py p 10 + ( ) 30 = 0 px py p ( ) + ( ) + ( ) = 1 px py p K13483_C00.indd 70 9/13/01 11:04:43 AM

31 Review of Fore, Stress, and Strain Tensors 71 Solving simultaneously, we obtain px = py = p = Diretion osines for σ p3 : The three independent equations beome: ( ) = 0 p3x p3y p ( ) 30 = 0 p3x p3y p3 ( ) + ( ) + ( ) = 1 p3x p3y p3 Solving simultaneously, we obtain: p3x = p3y = p3 = Plane Stress A stress tensor is always defined by six omponents of stress: three normal stress omponents and three shear stress omponents. However, in pratie a state of stress is often enountered in whih the magnitudes of three stress omponents in one oordinate diretion are known to be ero a priori. For example, suppose σ = τ x = τ y = 0, as shown in Figure.13a. As the three remaining nonero stress omponents (σ xx, σ yy, and τ xy ), all lie within the x y plane, suh a ondition is alled a state of plane stress. Plane stress onditions our most often beause of the geometry of the struture of interest. Speifially, the plane stress ondition usually exists in relatively thin, platelike strutures. Examples inlude the web of an I-beam, the body panel of an automobile, or the skin of an airplane fuselage. In these instanes the stresses indued normal to the plane of the struture are very small ompared with those indued within the plane of the struture. Hene, the small out-of-plane stresses are assumed to be ero, and attention is foused on the relatively high stress omponents ating within the plane of the struture. K13483_C00.indd 71 9/13/01 11:04:46 AM

32 7 Strutural Analysis of Polymeri Composite Materials (a) σ xx +x +y τ xy τ yx σ yy + (b) +x σ xx τ xy σ yy σ yy +y τ xy σ xx Figure.13 Stress elements subjeted to a state of plane stress. (a) 3-D stress element subjeted to a plane stress state (all stress omponents shown in a positive sense); (b) plane stress element drawn as a square rather than a ube (positive -axis out of the plane of the figure; all stress omponents shown in a positive sense). As laminated omposites are often used in the form of thin plates or shells, the plane stress assumption is widely appliable in omposite strutures and will be used throughout most of the analyses disussed in this book. As the out-of-plane stresses are negligibly small, for onveniene an infinitesimal stress element subjeted to plane stress will usually be drawn as a square rather than a ube, as shown in Figure.13b. Results disussed in earlier setions for general 3D state of stress will now be speialied for the plane stress ondition. It will be assumed that the nonero stresses lie in the x y plane (i.e., σ = τ x = τ y = 0). This allows the remaining omponents of stress to be written in the form of a olumn array, rather than a 3 3 array: σxx τxy 0 σ τxy σ yy 0 σ τ xx yy xy K13483_C00.indd 7 9/13/01 11:04:46 AM

33 Review of Fore, Stress, and Strain Tensors 73 Note that when a plane stress state is desribed, stress appears to be a first-order tensor, as (apparently) only three omponents of stress (σ xx, σ yy, and τ xy ) need be speified to desribe the state of stress. This is, of ourse, not the ase. Stress is a seond-order tensor in all instanes, and six omponents of stress must always be speified to define a state of stress. When we invoke the plane stress assumption, we have simply assumed a priori that the magnitude of three stress omponents (σ, τ x, and τ y ) are ero. Reall that either Equation.1a or Equation.1b governs the transformation of a stress tensor from one oordinate system to another. Equation.1b is repeated here for onveniene: σ σ σ σ σ σ σ σ σ x x x y x y x y y y x y = x x x y x y x y y y x y σxx σxy σx σ yx σ yy σ y σx σy σ x x y x x x y y y y x y (.1b) (repeated) When transformation of a plane stress tensor is onsidered, it will be assumed that the x y oordinate system is generated from the x y system by a rotation θ about the -axis. That is, the - and -axes are oinident, as shown in Figure.14. In this ase, the diretion osines are x x x y x y x y y y x y = os( θ) = os( 90 θ) = sin( θ) = os( 90 ) = 0 = os( 90 + θ) = sin( θ) = os( θ) = os( 90 ) = 0 = os( 90 ) = 0 = os( 90 ) = 0 = os( 0 ) = 1 If we now (a) substitute these diretion osines into Equation.1b, (b) label the shear stresses using the symbol using τ rather σ, K13483_C00.indd 73 9/13/01 11:04:47 AM

34 74 Strutural Analysis of Polymeri Composite Materials (a) +x σ xx τ xy σ yy σ yy +y τ xy σ xx (b) +x +θ +x σ x x σ y y τ x y +y τ x y +θ σ y y σ x x +y Figure.14 Transformation of a plane stress element from one oordinate system to another. (a) Plane stress element referened to the x y oordinate system; (b) plane stress element referened to the x y oordinate system, oriented θ-degrees ounter-lokwise from the x y oordinate system. and () note that σ = τ x = τ y = 0 by assumption, then Equation.1b beomes: σ τ τ τ σ τ τ τ σ x x x y x y x y y y x y osθ sinθ 0 σxx τxy 0 = sinθ osθ 0 τyx σ yy osθ sinθ 0 sinθ osθ K13483_C00.indd 74 9/13/01 11:04:48 AM

35 Review of Fore, Stress, and Strain Tensors 75 Completing the matrix multipliation indiated results in: σ τ τ τ σ τ τ τ σ x x x y x y x y y y x y os θσxx + sin θσ yy + osθsinθτxy osθsinθσ xx + osθsin θσ yy + (os θ sin θ) τxy 0 = osθsinθσ xx + osθsin θσ yy + (os θ sin θ) τxy sin θσxx + os θσ yy osθsinθτ xy (.18) As would be expeted, the out-of-plane stresses are ero: σ = τ x =τ y =0. The remaining stress omponents are σ = os ( θ) σ + sin ( θ) σ + os( θ)sin( θ) τ x x xx yy xy σ y y = sin ( θ) σ + os ( θ) σ os( θ)sin( θ) τ xx yy xy (.19) τ = os( θ)sin( θ) σ + os( θ)sin( θ) σ + [os ( θ) sin ( θ)] τ x y xx yy xy Equation.19 an be written using matrix notation as σ σ τ x x y y x y os ( θ) sin ( θ) os( θ)sin( θ) σ = sin ( θ) os ( θ) os( θ)sin( θ) σ os( θ)sin( θ) os( θ)sin( θ) os ( θ) sin ( θ) τ xx yy xy (.0) It should be kept in mind that these results are valid only for a state of plane stress. More preisely, Equations.19 and.0 represent stress transformations within the x y plane, and are only valid if the -axis is a prinipal stress axis. The 3 3 array that appears in Equation.0 is alled the transformation matrix, and is abbreviated as [T]: os ( θ) sin ( θ) os( θ)sin( θ) [ T ] = sin ( θ) os ( θ) os( θ)sin( θ) os( θ)sin( θ) os( θ)sin( θ) os ( θ) sin ( θ) (.1) The stress invariants (given by Equation.14 or.15) are onsiderably simplified in the ase of plane stress. Sine by definition σ = τ x = τ y = 0, the stress invariants beome: K13483_C00.indd 75 9/13/01 11:04:49 AM