MEG 74 Energ and Variational Methods in Mechanics I Brendan J. O Toole, Ph.D. Associate Professor of Mechanical Engineering Howard R. Hghes College of Engineering Universit of Nevada Las Vegas TBE B- (7) 895-3885 bj@me.nlv.ed Chapter : Differential Form of Basic Eqations -
Part Vectors, Tensors, and Matrices -
Scalar, Vector & Tensor Eamples Speed (Scalar or th order Tensor) It has magnitde onl and is represented b a single vale. Velocit (Vector or st order tensor) It has magnitde and direction, 3 components are needed to fll define a vector, e.g.: v (v, v, v ). Stress ( nd order tensor) Stress at a point has different magnitdes in different directions. Nine components are needed to define stress. σ σ σ σ σ σ σ σ σ σ -3
Sbscript Notation, σ, σ First Sbscript Signifies the ais to which the otward normal of the face is parallel Second Sbscript Indicates the ais to which the stress is parallel σ is a stress acting on the -face in the -direction which implies that it is a shear stress. σ is a stress acting on the -face in the -direction which implies that it is a normal stress. The second sbscript is sometimes omitted for normal stresses. If onl one sbscript is given, assme that it is a normal stress. -4
Sign Convention for Stresses on a Control Element of an Elastic Bod σ τ τ τ τ σ τ τ σ σ τ τ τ τ σ τ τ σ Positive Stress Directions On Positive Faces Positive Stress Directions On Negative Faces -5
Sign Convention For Stresses Positive Faces of Control Element A positive face has an otward normal vector in the positive direction of an ais. Stresses are positive when the are in the positive direction of an ais Negative Faces of Control Element A negative face has an otward normal vector in the negative direction of an ais. Stresses are positive when the are in the negative direction of an ais -6
Sbscript Notation for Stresses Normal Stresses (σ or σ ) Shear Stresses (τ or σ ) The Greek smbol σ is sall sed to represent a normal stress and τ is sed to represent a shear stress. When working with vector or tensor eqations it is necessar to se one smbol to represent all tpes of stresses. In general, σ is sed to represent all stresses and the sbscripts designate whether it is a normal stress or shear stress. -7
Tensors & Indicial Notation Scalars and vectors are sbsets of the general definition of a tensor qantit. Higher order tensors have more components. Einstein developed a shorthand notation to simplif writing tensor eqations. With indicial notation, the sbscripts i and j are sed to represent all aes (,, and ). For eample, σ ij is a shorthand version of writing all nine components of the entire stress tensor. -8
Eamples Using Indicial Notation Vectors ( st order tensors) The vector (v, v, v ) can be written as v i or v j, the sbscript represents all three components. nd order tensors (Stress and Strain) The stress (or strain) at a point is represented b nine different vectors. σ ij is the shorthand method for writing all nine components where i,, and j,,. ij where, and are normal strains and the rest are shear strains; e.g: γ /. -9
Smmation Convention Component Form of Vector A A a e a e a 3 e 3, where e, e, and e 3 are base vectors Smmation Notation for Vector A 3 j j A a e j a e j Shorthand Notation for Vector A j k m A a e a e a e j k m j -
Second Order Tensor Component Form Inditial Notation Form Φ φ e e φ e e φ 3 e e 3 φ e e φ e e φ 3 e e 3 φ 3 e 3 e φ 3 e 3 e φ 33 e 3 e 3 Φ φ ij e i e j Unit Second Order Tensor, I I δ ij e i e j δ ij if if i i j j -
Dot Prodct Between Second Order Tensors (Doble Dot Prodct) Similar to single dot prodct, encontered periodicall in mechanics derivations Φ : Ψ (φ ij e i e j ) : (ψ mn e m e n ) φ ij ψ mn (e j. e m ) (e i. e n ) φ ij ψ mn δ jm δ in φ im ψ mn δ in φ nm ψ mn φ ij ψ ji -
Matrices and Matri Operations The components of a tensor can be written down in a rectanglar arra (or matri). Sqare matri: n n (has order n) Diagonal Elements: elements with the same row and colmn nmber Diagonal Matri: Onl non-ero elements are along the diagonal Identit Matri [I]: Diagonal matri with all non-ero elements -3
Matrices and Matri Operations Trace of a Matri: Sm of the diagonal elements Smmetric Matri: [A] [A] T Skew Smmetric Matri: [A] T - [A] Review on or own: Matri addition, mltiplication Calclating determinants, Cramer s Rle Matri Transformations -4
Part Theor of Elasticit -5
Governing Eqations For Elastic Bodies Kinematics Strain-Displacement Eqations Thermodnamics st and nd Laws of Thermodnamics (Compatibilit Eqations) Constittive Eqations Stress-Strain Relations Kinetics Conservation of momenta -6
Kinematics Kinematic eqations are sed to describe the motion and deformation of deformable bodies Lagrangian (or referential) Description The motion of the bod, B, is described relative to a reference configration C R (sall the nstressed state of the bod). The crrent coordinates (,, ) are epressed in terms of the reference coordinates (,, ), which are sometimes called the material coordinates, and time, t. Spatial (or Elerian) Description The motion is referred to the crrent configration of the bod. The spatial description focses on a region of space, instead of on a given bod. Elerian descriptions are sed most often in flid mechanics and is sometimes referred to as a control volme approach. -7
Lagrangian Strain, C O r Q r P Q ds P r Q P r P Q,, C Q ds P A bod ndergoes deformation, P and Q represent particles which occp the initial positions defined b r P & r Q. The final position of the two particles are represented b r P & r Q. The particles are separated b the infinitesimal distance, ds, in the ndeformed state and ds, in the deformed state. -8
Strains The strain tensor, є, is defined in terms of the change in distance between the two particles while moving from the ndeformed state to the deformed state. It can be shown that these distances are related to the displacements, P and Q. є is known as the Green-Lagrange strain tensor ik ( ) i, k k, i l, i l, k -9
- Cartesian Components of Strain (Green Lagrange Strain Tensor) w w v v w v w w v v w w w v v v w v w w v v w v displacement in the direction v displacement in the direction w displacement in the direction
Infinitesimal Strain Green-Lagrange strain tensor: ik ( ) i, k k, i l, i l, k For small strains, the higher order terms are negligible The redced form of the strain tensor is called the infinitesimal or Cach strain tensor: ik ( ) i, k k, i -
- Cartesian Components of Strain (Infinitesimal or Cach Strain Tensor) w w w v w v w v,,,,,,,,, γ γ γ displacement in the direction v displacement in the direction w displacement in the direction
-3 Matri Form of the Cach Strain Tensor D γ γ γ D is the differential operator matri
Compatibilit Eqations Assming we have epressions for the displacements and these epressions are differentiable, we can determine the strains sing the previosl defined eqations. However, if the si strains are known and we wish to determine the three displacements, o mst be sre that a niqe soltion eists. The compatibilit conditions ensre that a niqe set of displacements are fond and that there are no kinks, tears, gaps, or overlaps. Some of the strain components mst be related to ensre compatibilit. -4
-5 Compatibilit Relations The compatibilit relations are defined b considering derivatives of individal strain components. For eample, take the partial derivative of γ with respect to and : v v γ γ γ v γ Since Rearranging order of partial derivatives Sbstitting definitions of strain This is strain compatibilit eqation. There are 6 of these for a general 3-D problem.
Compatibilit Relations The si compatibilit relations can be written sing indicial notation as: ij mn im m n i For -D problems, this redces to a single eqation: j j n i jn m See Eq..8 for the si eqations. -6
Constittive Relations (Stress - Strain Eqations) The stress-strain eqations or constittive relations are a good eample of the seflness of the indicial notation. The most basic constittive relation that we first learn in Mechanics of Materials is the -D Hookes Law eqation: σ E, Stress eqals Yong's modls times strain. Later, we learned that if there is strain in more than one direction, the stress will be a fnction of all strains. -7
General 3-D Constittive Relations The first two (of nine) stress eqations can be written as shown below. Each of the nine stress components is a fnction of all nine strain components. Each Q variable is a different fnction of material properties. σ Q Q Q Q Q Q Q Q Q σ Q Q Q Q Q Q Q Q Q -8
General 3-D Constittive Relations Both the shorthand and fll matri versions of the 3-D constittive eqations are shown below. σ ij Q ijkl kl σ σ σ σ σ σ σ σ σ where i, j, k and l represent, and. Sometimes the nmbers, and 3 are sbstitted for, and. Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q -9
Simplifing the Constittive Relations The general constittive eqations have 8 elastic constants. Lckil, the nmber of constants is redced for most practical materials. Both the stress and strain tensors are smmetric. σ ij σ ji and ij ji This redces the nmber of independent stresses and strains to 6. -3
Constittive Relations Since there are onl si independent stresses and strains, another shorthand notation is introdced sing nmeric sbscripts. Let σ σ and σ σ and σ 3 σ and 3 σ 4 σ σ and 4 σ 5 σ σ and 5 σ 6 σ σ and 6-3
Constittive Relations This redces the nmber of independent material constants to 36. σ σ σ3 σ4 σ5 σ6 Q Q Q3 Q4 Q5 Q6 Q Q Q3 Q4 Q5 Q6 Q3 Q3 Q33 Q34 Q35 Q36 Q4 Q4 Q43 Q44 Q45 Q46 Q5 Q5 Q53 Q54 Q55 Q56 Q6 Q6 Q63 Q64 Q65 Q66 A thermodnamics proof can be sed to show that the Q matri itself is smmetric, or Q ij Q ji. This redces the nmber of independent constants to. 3 4 5 6-3
Anisotrop and Material Smmetr General Anisotrop A material with different material properties in all directions ehibits general anisotrop. independent elastic constants are reqired to define the stress-strain relationship for this tpe of material. Orthotropic Material Has 3 mtall orthogonal planes of elastic smmetr e.g.: A material with the same properties in the and - directions has elastic smmetr abot the - plane. Most composite materials ehibit elastic smmetr abot three planes Reqires 9 independent elastic constants -33
Orthotropic Stress-Strain Eqations σ σ σ3 σ4 σ 5 σ6 or σ i Q ij j, where Q ij is know as the stiffness matri. Inverting the matri eqation ields: 3 4 5 6 Q Q Q3 Q Q Q3 Q3 Q3 Q33 Q44 Q 55 Q66 S S S3 S S S3 S3 S3 S33 S44 S 55 S66 or i S ij σ j, where S ij is know as the compliance matri. [S] [Q] - σ σ σ3 σ4 σ 5 σ6 3 4 5 6-34
Compliance Matri The compliance matri is easier to define in terms of elastic properties. 3 4 5 6 S S S3 S S S3 S3 S3 S33 S44 S 55 S66 σ σ σ3 σ4 σ 5 σ6 S E S 44 G 3 S E S 55 G 3 S 33 E 33 S 66 G S S ν E ν E S 3 S 3 ν 3 E ν 3 E 33 S 3 S 3 ν 3 E ν 3 E 33-35
Stiffness Matri TheQ ij components are fond b inverting S ij Q E ( ν 3 ν 3 ) Q 44 G 3 Q E ( ν 3 ν 3 ) Q 55 G 3 Q 33 E 33 ( ν ν ) Q 66 G Q E ( ν ν 3 ν 3 ) E ( ν ν 3 ν 3 ) Q 3 E ( ν 3 ν ν 3 ) E 33 ( ν 3 ν ν 3 ) Q 3 E ( ν 3 ν ν 3 ) E 33 ( ν 3 ν ν 3 ) ν ν ν 3 ν 3 ν 3 ν 3 ν ν 3 ν 3-36
-37 Isotropic Stress-Strain Relations ( )( ) E γ γ γ ν ν ν ν ν ν ν ν ν ν ν ν τ τ τ σ σ σ ν ν