COORDINATE TRANSFORMATION When [σ]xyz is given, what would be the components in a different coordinate system x′y′z′ (i.e., [σ]x′y′z′)? Unit vectors in x′y′z′-coordinates: b1 = {1, 0, 0}T in x′y′z′ coordinates, while in xyz coordinates the rotational transformation matrix Stress does not rotate. The coordinates rotate

COORDINATE TRANSFORMATION cont. [N] transforms a vector in the x′y′z′ coordinates to the xyz coordinates. [N]T transforms a vector in the xyz coordinates to the x′y′z′ coordinates. Consider bx′y′z′ = {1, 0, 0}T: Stress transformation: Using stress vectors, By multiplying [N]T the stress vectors can be represented in the x′y′z′ coordinates The first [N] transforms the plane; the second transforms the force. See example in notes page >> Sigma=[ 3 1 1; 1 0 2; 1 2 0] % The stress matrix of Example 1.4 in textbook Sigma = 3 1 1 1 0 2 1 2 0 >> [V,D]=eig(Sigma) % Calculate the principal directions (V) and principal stresses V = % V gives the unit vectors of the principal system -0.0000 0.5774 -0.8165 0.7071 -0.5774 -0.4082 -0.7071 -0.5774 -0.4082 D = -2.0000 0 0 0 1.0000 0 0 0 4.0000 >>Sigprinc= V'*A*V % So we can use it as the transformation matrix N = %and when we apply it to the stress matrix we get the stress matrix in -2.0000 -0.0000 0 % the principal system -0.0000 1.0000 0.0000 0 0.0000 4.0000

MAXIMUM SHEAR STRESS Important in failure criteria of the material Mohr’s circle maximum shear stress Normal stress at max shear stress plane Example in the notes page We now take the stress matrix in the principal system D = -2.0000 0 0 0 1.0000 0 0 0 4.0000 >> a=1/sqrt(2) % and transform it to a system where the x axis is at 45 degrees a = % between the direction of the minimum and maximum principal 0.7071 % stress >> N=[a 0 -a; 0 1 0; a 0 a]' N = 0.7071 0 0.7071 -0.7071 0 0.7071 >> Sigshear=N'*D*N % As can be seen below the tau13 is now the maximum Sigshear = 1.0000 0 -3.0000 -3.0000 0 1.0000

Quiz-like questions The stress matrix is given as What is the maximum shear stress? What is the transformation matrix N for a coordinate system rotated 45 degrees about z axis, so that the positive x’ axis is in between the x and y axes? What is the transformed stress matrix. Answers in notes page. Since the stress matrix is diagonal, there are no shear stresses, which means that we are in the principal coordinate system and the principal stresses are Each axis represented by the cosines of its angles with the old x,y,z axes. The x’ axis makes 45 degrees with x and y and 90 degrees with z The y’ axis makes 135 degrees with x, 45 with y and 90 with z The z’ axis is the same as z, so the angles are 90,90, and 90 degrees. So the matrix of direction cosines is >> sdiag=[-2 0 1]; sigma=diag(sdiag) sigma = -2 0 0 0 0 0 0 0 1 >> a=sqrt(2)/2; N=[a a 0; -a a 0; 0 0 1]; >> sigtrans=N'*sigma*N sigtrans = -1.0000 -1.0000 0 0 0 1.0000

What Stresses Could Be Used in Failure and Design Criteria? Must be independent of the coordinate system. Stress Invariants Principal Stresses Maximum Shear Stress

STRAIN Strain: a quantitative measure of deformation Normal strain: change in length of a line segment Shear strain: change in angle between two perpendicular line segments Displacement of P = (u, v, w) Displacement of Q & R P(x,y,z) Q R P'(x+u,y+v,z+w) Q' R' x y z Dx Dy

Textbook has different, but more rigorous derivations NORMAL STRAIN Strain is defined as the elongation per unit length Tensile (normal) strains in x- and y-directions Strain is a dimensionless quantity. Positive for elongation and negative for compression P Dx Dux Dy Duy Textbook has different, but more rigorous derivations

SHEAR STRAIN Shear strain is the tangent of the change in angle between two originally perpendicular axes Shear strain (change of angle) Positive when the angle between two positive (or two negative) faces is reduced and negative when the angle is increased. Valid for small deformation P Dux Duy q2 q1 p/2 – g12 Dy Dx

STRAIN MATRIX Strain matrix and strain vector Normal component: Why do we bother with both Coordinate transformation: Principal strain: ε1 ≥ ε2 ≥ ε3 Maximum shear strain: Will the principal direction of strain be the same as that of stress?

Quiz-like questions The only non-zero strain component is gxz=0.05 What is the physical meaning of this strain component? What are the principal strains? What is the maximum engineering shear strain gmax ? Answers in notes page The physical meaning of this shear strains is that the angle between lines that were originally aligned with the x and z axes are reduced from p/2 to p/2-0.05 To find the principal strains we need to use exz = 0.5gxz eps=[0 0 0.025; 0 0 0; 0.025 0 0]; principal=(eig(eps))' principal = -0.0250 0 0.0250 gmax =0.025-(-0.025)=0.05

STRESS VS STRAIN [] is a symmetric 3×3 matrix Normal stress in the direction n is Normal strain in the direction n is Transformation of stress Transformation of strain Three mutually perpendicular principal directions and principal stresses can be computed as eigenvalues and eigenvectors of the stress matrix as Three mutually perpendicular principal directions and principal strains can be computed as eigenvalues and eigenvectors of the strain matrix as