HOMEWORK From Dieter 2-7 Module #4 Fundamentals of strain The strain deviator Mohr s circle for strain READING LIST DIETER: Ch. 2, Pages 38-46 Pages 11-12 in Hosford Ch. 6 in Ne
Strain When a solid is subjected to a load, parts of the solid are displaced from their original positions. Think of it like this; the atoms making up the solid are displaced from their original positions. C Load B A B C A O O Load This displacement of points or particles under an applied stress is termed strain.
Total Displacement = -v u + + v v Translation Rotation Pure Shear
Where to begin Consider a point A in a solid located at position,,z. z w A (,,z) u u A A (+u,+v,z+w) v Appl force to the bod and point A (,,z) is displaced to A (+u,+v,z+w).
Displacement of points Displacement vector: u A = f(u,v,w) where u, v, and w are units of translation along the,, z aes. Solids are composed of man particles. If u A is constant for all particles, no deformation occurs (onl translation). If u A varies from particle to particle, i.e., u i = f( i ), the solid deforms. z A (,,z) u A A (+u,+v,z+w) Displacement = Translation + Rotation + Shear
1-D D Linear Strain A B d A B F 0 1 2 3 4 5 Points A and B are displaced from their original positions The amount of displacement is a function of. Point B moves farther than Point A. Let distance A A = u. Thus, distance B B = u+(δu/δ)d = u+(u/)d.
1-D D Linear Strain cont d. A B d A B F Strain is defined b the following relationship: e u d d d L AB AB u u L AB d Integrating ields the displacement. u u e u o rigid bod translation, which we can subtract, ielding: u e o
Generalization to 3-D3 Displacement is related to the initial coordinates of the point. u ee ez z v ee ez zu w ezez ezzz i ij j Normal / Linear Strains: e u z w A (,,z) u A A (+u,+v,z+w) v u v w e, e, ezz z u v w e, e, ezz z If we orient the sstem such that the load is applied parallel to the -ais. The variables u, v, and w are displacements parallel to the,, and z aes.
Shear Strains in 2-D 2 D and 3-D3 Consider a square or cubic element that is distorted b shear. C D D C u A v B B Incremental displacement in -direction = u. Incremental displacement in -direction = v. Incremental displacement in z-direction = w.
2D Displacement of AD increases with distance along the -ais resulting in an angular distortion of -ais. C e DD u u or h DA shear distortion of the -ais in the -direction An analogous event occurs along the -ais. D A e D u v C B B BB v v or h AB shear distortion of the -ais in the -direction
Strain in 3-D3 The displacement strain is defined b nine strain components: e, e, e z, e, e, e z, e zz, e z, e z The strains on the negative faces are equal to satisf the requirements for equilibrium. z e z e z e zz e z e z e Notation is similar to stress; subscripts reversed: e ij : i = direction of displacement j = plane on which strain acts e e e Convention (+)ive when both i & j are (+)ive (+)ive when both i & j are (-)ive (-)ive when both i & j are opposite Tension: e ij = positive Compression: e ij = negative
3D Displacement Strain Matri u u u u u u z z e e ez v v v v v v eij e e ez z z ez ez ezz w w w w w w z z The displacement strain matri. Can produce pure shear strain and rigid-bod rotation.
u e = e u e = -e u e = u/ e = 0 v -v (1) Pure Shear w/o Rotation (2) Rotation (3) Simple Shear We need to break the displacement matri into strain and rotational components. We can decompose the total strain matri into smmetric and anti-smmetric components.
Decomposition of Strain e ij ij ij 1 1 ij ji ij 2 2 e e e e ji 1 2 u i j u j 1 u u i 2 Smmetric i j Anti-smmetric i j Shear Rotation
Displacement strain [matri] e e e e e e e e e z z z z zz = = Shear strain [tensor] + ε ij 1 1 e e e ez ez 2 2 z 1 1 z e e e ez ez 2 2 z z zz 1 1 ez ez ez ez e zz 2 2 + Rotation [tensor] 1 1 0 e e ez ez 2 2 z 1 1 ij z e e 0 ez ez 2 2 z z zz 1 1 ez ez ez ez 0 2 2
Shear Strain Total angular change from a right angle. u e e 2 ( ij 0) e = e ij 2 (engineering shear strain) ij (1) Pure Shear w/o Rotation v z z u v w u z w v z
Transformation of Strains Equations for strain, analogous to those for stress, can be written b substituting for and /2 for. l m n 2lm 2mn 2nl normal 2 2 2 zz z z l m n lm mn nl normal 2 2 2 zz z z We can also define a coordinate sstem where there will be no shear strains. These will be principal aes
1 - - - - 4 1 1 4 4 or zz zz zz z z 3 2 2 2 2 2 2 2 - zz z z - z z zz 0 -I I -I 0 3 2 1 2 3 The directions in which the principal strains act are determined b substituting 1, 2, and 3, each for in: ( - )2l m zn 0 l ( - )2m zn 0 l m( - )2n 0 z z zz and then solving the resulting equations simultaneousl for l, m, and n (using the relationship l 2 + m 2 + n 2 = 1). (a) Substitute 1 for in & solve; (b) Substitute 2 for in & solve; (c) Substitute 3 for in & solve.
Equations for Principal Shearing Strains 1 2 3 ma 2 1 3 3 1 2 Deformation of a solid involves a combination of volume change and shape change. We can separate strain into hdrostatic (volume change) and deviatoric (shape change) components.
Hdrostatic Component Volume = dddz z d d dz Volume of strained element 1 1 1 = zz dddz The volume strain is: 1 1 1 dddz dddz zz dddz 1 1 1 1 zz If we neglect the products of strains (i.e., ε ii ε jj ), this becomes: zz which is equal to the first invariant of the strain tensor
The hdrostatic component of strain, i.e., the mean strain, is: zz ii mean 3 3 3 The mean strain does not induce shape change. It causes volume change. It is the hdrostatic component. The part that causes shape change is called the strain deviator. We get the strain deviator b subtracting the mean strain from the normal strain components. mean z ij mean z z z zz mean
The Strain Deviator mean z ij mean z z z zz mean 2 zz 3 z 2 zz 3 z z z 2 zz 3 ij ij m ij ij ij 3 3
Mohr s s Circle for Strain +γ/2 CW Allows us to determine the magnitude and directions of the principal strains. ma /2 H(, /2) Intersection with the -ais is min = 2 ( 2, ) C( average, 0) 2 ( 1, ) ε V(, - /2) CCW Intersection with the -ais is ma = 1
MAXIMUM & MINUMUM PRINCIPAL STRAINS IN 2-D STATE 2 2 ma 1 min 2 2 2 2 2 2 ma 3 tan 2 normal tan 2 shear
Strain Measurement Strain can be measured using a strain gauge. When an object is deformed, the wires in the strain gate are strained which changes their electrical resistance, which is proportional to strain. Strain gauges make onl direct readings of linear strain. Shear strain must be determined indirectl. 45 60 45 45 60 120 60 60 Rectangular Delta
State of stress at a point: z 11 12 13 z 21 22 23 z z zz 31 32 33 11 12 13 12 22 23 13 23 33 State of strain at a point: 11 12 13 11 12 13 21 22 23 12 22 23 31 32 33 13 23 33 There are man different sstems of notation. BE WARY!
Matri Notation We often replace the indices with matri notation for simplicit 11 1 11 2 zz 33 3 z 23 4 z 13 5 12 6 11 12 13 1 6 5 22 23 2 4 33 3 This will be particularl important when we discuss higher order tensors and tensor relationships (i.e., elastic properties)
General forms for stress and strain in matri notation 1 6 5 6 2 4 5 4 3 6 5 1 2 2 6 4 2 2 2 5 5 3 2 2 Special definitions NOTE ; ; 1 11 2 22 3 33 2 4 23 23 2 5 13 13 2 6 12 12