� � � MIT 3.016 Fall 2005 � W.C Carter Lecture 10 c55 Oct. 07 2005: Lecture 10: Real Eigenvalue Systems; Transformations to Eigenbasis Reading: Kreyszig Sections: 7.4 (pp:385–89) , 7.5 (pp:392–96) § §Similarity Transformations A matrix has been discussed as a linear operation on vectors. The matrix itself is defined in terms of the coordinate system of the vectors that it operates on—and that of the vectors it returns. In many applications, the coordinate system (or laboratory) frame of the vector that gets operated on is the same as the vector gets returned. This is the case for almost all physical properties. For example: In an electronical conductor, local current density, J�, is linearly related to the local • electric field �E: ρJ�= E (10-1) • In a thermal conductor, local heat current density is linearly related to the gradient in temperature: k�T = j�Q (10-2) In diamagnetic and paramagnetic materials, the local magnetization, �B is related to the • applied field, �H: µH = B (10-3) • D, is related to the applied electric In dielectric materials, the local total polarization, � field: κ� �E = D = κoE + �(10-4)�P When �x and �y are vectors representing a physical quantity in Cartesian space (such as force �E, orientation of a plane ˆF , electric field �n, current �j, etc.) they represent something physical. They don’t change if we decide to use a different space in which to represent them (such as, exchanging x for y, y for z, z for x; or, if we decide to represent length in nanometers instead of inches, or if we simply decide to rotate the system that describes the vectors. The representation of the vectors themselves may change, but they stand for the same thing.MIT 3.016 Fall 2005 � W.C Carter Lecture 10 c56 One interpretation of the operation A� x has b een described as geometric transformation on the vector � x . For the case of orthogonal matrices A orth, geometrical transformations take the forms of rotation, reflection, and/or inversion. Suppose we have some physical relation between two physical vectors in some coordinate system, for instance, the general form of Ohm’s law is: J� =σE� ⎛⎝ ⎞⎠ = ⎛⎝ ⎛⎝ ⎞⎠ ⎞⎠ (10-5) J σ σ σ Ex xx xy xz x J σ σ σ Ey xy yy yz y J σ σ σ Ez xz yz zz z The matrix (actually it is better to call it a rank-2 tensor) σ is a physical quantity relating the amount of current that flows (in a direction) proportional to the applied electric field (perhaps in a different direction). σ is the “conductivity tensor” for a particular material. The physical law in Eq. 10-5 can be expressed as an inverse relationship: E� =ρ�j ⎛⎝ ⎞⎠ = ⎛⎝ ⎛⎝ ⎞⎠ ⎞⎠ (10-6) E ρ ρ ρ jx xx xy xz x E ρ ρ ρ jy xy yy yz y E ρ ρ ρ jz xz yz zz z where the resistivity tensor ρ = σ −1 . What happens if we decide to use a new coordinate system (i.e., one that is rotated, reflected, or inverted) to describe the relationship expressed by Ohm’s law? The two vectors must transform from the “old” to the “new” coordinates by: old→new E�old→new j�new A orth old = j� newA orth old = E� newE� new = E�old new old j� old A orth →old A orth →new = j�(10-7) Where is simple proof will show that: old new newA orth →=A orth →old −1 new old new−1 →A orth →old =A orth (10-8) new old newT →A orth →old =A orth new old newT →A orth →old =A orth where the last two relations follow from the special properties of orthogonal matrices.MIT 3.016 Fall 2005 � W.C Carter Lecture 10 c57 How does the physical law expressed by Eq. 10-5 change in a new coordinate system? old = χ old E�old in old coordinate system: j� (10-9) newin new coordinate system: j� new = χ new E� To find the relationship between χ old and χ new : For the first equation in 10-9, using the transformations in Eqs. 10-7: new old j�A orth →new = χ oldA newnew orth →old E� (10-10) and for the second equation in 10-9: old new j�→old = χ new A old→new E�old (10-11)A orth orth Left-multiplying by the inverse orthogonal transformations: old new new old j� new old→new χ oldA newnewA orth →A orth →= A orth orth →old E� new old new j� new new A old→new E�(10-12) old →old = A orth →old χA orth →old A orth orth Because the transformation matrices are inverses, the following relationship between similar matrices in the old and new coordinate systems is: χ old old→new new A new old = A orth χ orth →(10-13)newχ new = A orth →old χ oldA old new→orth The χ old is said to be similar to χ new and the double multiplication operation in Eq. 10-13 is called a similarity transformation. cqkcqStresses and Strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stresses and strains are rank-2 tensors that characterize the mechanical state of a material. A spring is an example of a one-dimensional material—it resists or exerts force in one direction only. A volume of material can exert forces in all three directions simultaneously— and the forces need not be the same in all directions. A volume of material can also be “squeezed” in many different ways: it can be squeezed along any one of the axis or it can be subjected to squeezing (or smeared) around any of the axes2 2Consider a blob of modeling clay—you can deform it by placing between your thumbs and one opposed finger; you can deform it by simultaneously squeezing with two sets of opposable digits; you can “smear” it by pushing and pulling in opposite directions. These are examples of uniaxial, biaxial, and shear stress.xyzσyxσzxσxxσyyσzyσxyσyzσzzσxzσ21σ31σ11σ22σ32σ12σ23σ33σ13�MIT 3.016 Fall 2005 � W.C Carter Lecture 10 c58 All the ways that a force can be applied to small element of material are now described. A force divided by an area is a stress—think of it the areal density of force. F i F x F�ˆi σ ij = (i.e., σ xz = = σ xz = · ) (10-14)A j A z A�k ˆ· A j is a plane with its normal in the ˆj -direction (or the projection of the area of a plane A�in the direction parallel to ˆj ) Figure 10-1: Illustration of stress on an oriented volume element. σ xx