Mechanics of Materials Lab
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1 Mechanics of Materials Lab Lecture 9 Strain and lasticity Textbook: Mechanical Behavior of Materials Sec. 6.6, 5.3, 5.4 Jiangyu Li Jiangyu Li,
2 Prof. M.. Tuttle Strain: Fundamental Definitions "Strain" is a measure of the deformation of a solid body There are two "types" of strain; normal strain (ε) and shear strain (γ) ε = (change in length) (original length) ; units = in / in, m / m, etc γ = (change in angle ) ; units = radians 28
3 Prof. M.. Tuttle Strain Within a Plane abc = π/2 radians abc < π/2 radians +y +y a ly a c b lx Original Shape +x ly + ly b lx + lx c Deformed Shape εxx = lx/lx +x εyy = ly/ly γxy = ( abc) 29
4 Strain Sign Convention A positive (tensile) normal strain is associated with an increase in length A negative (compressive) normal strain is associated with a decrease in length A shear strain is positive if the angle between two positive faces (or two negative faces) decreases +y +y +x +x All Strains Positive ε xx Positive ε and γ Negative yy xy 32
5 Prof. M.. Tuttle 3-Dimensional Strain States In the most general case, six components of strain exist "at a point": ε xx, ε yy, ε zz, γ xy, γ xz, γ yz Strain is a 2 nd -order tensor; the numerical values of the individual strain components which define the "state of strain" depend on the coordinate system used This presentation will primarily involve strains which exist within a single plane 29
6 Prof. M.. Tuttle Strain Within a Plane We are often interested in the strains induced within a single plane; specifically, we are interested in two distinct conditions: "Plane stress", in which all non-zero stresses lie within a plane. The plane stress condition induces four strain components: ε xx, ε yy, ε zz, and γ xy "Plane strain", in which all non-zero strain components lies within a plane. By definition then, the plane strain condition involves three strain components (only): ε xx, ε yy, and γ xy 30
7 Prof. M.. Tuttle Strain Transformations Given strain components in the x-y coordinate system (ε xx, ε yy, γ xy ), what are the corresponding strain components in the x'-y' coordinate system? ly + ly +y? +y' +x' θ lx + lx εxx = lx/lx εyy = ly/ly γxy = ( abc) +x εx'x' =? εy'y' =? γx'y' =? 35
8 Prof. M.. Tuttle Strain Transformation quations Based strictly on geometry, it can be shown: ε x' x' = ε xx + ε yy 2 ε y' y' = ε xx + ε yy 2 + ε xx ε yy 2 ε xx ε yy 2 cos2θ + γ xy 2 sin2θ cos2θ γ xy 2 sin2θ γ x'y' 2 = ε ε xx yy 2 sin 2θ + γ xy 2 cos2θ 36
9 Prof. M.. Tuttle "Constitutive Models" The structural engineer is typically interested in the state of stress induced in a structure during service The state of stress cannot be measured directly... The state of strain can be measured directly... Hence, we must develop a relation between stress and strain...this relationship is called a "constitutive model," and the most common is "Hooke's Law" 39
10 Prof. M.. Tuttle 3-D Form of Hooke's Law: Apply Principle of Superposition The total strain ε xx caused by all stresses applied simultaneously is determined by "adding up" the strain ε xx caused by each individual stress component ε xx = σ xx or + νσ yy + νσ zz + [ 0 ] + [ 0] + [ 0] ε xx = 1 [ σ xx νσ yy νσ ] zz 51
11 Prof. M.. Tuttle Hooke's Law: 3-Dimensional Stress States Following this procedure for all six strain components: ε xx = 1 [ σ xx νσ yy νσ ] zz γ yz = 2(1+ ν)τ yz ε yy = 1 [ νσ xx + σ yy νσ ] zz γ zx = 2(1+ ν)τ zx ε zz = 1 [ νσ xx νσ yy + σ ] zz γ xy = 2(1+ ν)τ xy 52
12 Prof. M.. Tuttle Hooke's Law: 3-Dimensional Stress States σ xx = σ yy = [ (1+ ν)(1 2ν) (1 v)ε xx + νε yy + νε zz ] τ yz = γ yz 2(1+ ν) [ (1+ ν)(1 2ν) vε xx + (1 v)ε yy + νε zz ] τ zx = γ zx 2(1 + ν) σ zz = (1+ ν)(1 2ν) vε xx + νε yy + (1 v)ε zz [ ] τ xy = γ xy 2(1+ ν) 54
13 Prof. M.. Tuttle Hooke's Law: Plane Stress For thin components (e.g., web or flange of an I-beam, automobile door panel, airplane "skin", etc) the "in-plane" stresses are much higher than "out-of-plane" stresses: (σ xx,σ yy,τ xy ) >> (σ zz, τ xz, τ yz ) It is convenient to assume the out-of-plane stresses equal zero...this is called a state of "plane stress" +y +z +x 57
14 Prof. M.. Tuttle Hooke's Law: Plane Stress ε xx = 1 (σ xx νσ yy ) σ yy τ xy σ xx ε yy = 1 (σ yy νσ xx ) ε zz = ν (σ xx + σ yy ) γ xy = τ xy G = 2(1+ ν)τ xy γ xz = γ yz = 0 58
15 Prof. M.. Tuttle Hooke's Law: Plane Stress σ xx = (1- ν 2 ) (ε xx + νε yy ) σ yy τ xy σ xx σ yy = (1- ν 2 ) (ε yy + νε xx ) τ xy = Gγ xy = 2(1+ ν) γ xy σ zz = τ xz = τ yz = 0 59
16 Prof. M.. Tuttle Hooke's Law: Plane Strain For thick or very long components (e.g., thick-walled pressure vessels, buried pipe, etc) the "in-plane" strains are much higher than "out-of-plane" strains: (ε xx,ε yy,γ xy ) >> (ε zz,γ xz,γ yz ) It is convenient to assume the out-of-plane strains equal zero...this is called a state of "plane strain" +y +z +x 62
17 Prof. M.. Tuttle Hooke's Law: Plane Strain σ xx = (1 + ν)(1 2ν) (1 ν)ε xx + νε yy [ ] σ yy τ xy σ yy = (1+ ν)(1 2ν) νε xx + (1 ν)ε yy [ ] σ xx σ zz = ν (1+ ν)(1 2ν) ε xx + ε yy [ ] = ν σ xx + σ yy [ ] τ xy = Gγ xy = τ xz = τ yz = 0 γ xy 2(1+ ν) 63
18 Hooke's Law Plane Strain σ yy τ xy σ xx ε xx = 1 ν2 ν σ xx 1 ν ε yy = 1 ν 2 ν σ yy 1 ν 2(1+ ν) γ xy = τ xy ε zz = γ xz = γ yz = 0 σ yy σ xx...and: σ zz = ν[ σ xx + σ yy ] 66
19 Prof. M.. Tuttle Hooke's Law: Uniaxial Stress Truss members are designed to carry axial loads only (i.e., uniaxial stress) In this case we are interested in the axial strain only (even though transverse strains are also induced) 67
20 Prof. M.. Tuttle Hooke's Law: Uniaxial Stress For: σ yy = σ zz = τ yz = τ xz = τ zy = 0, "Hooke's Law" becomes: σ xx = ε xx "Shorthand" notation: { σ} = [ D] { ε} where [ D] = [ ] 68
21 Jiangyu Li, Orthotropic Materials = xy zx yz z y x xy zx yz z y yz x xz z zy y x xy z zx y yx x xy zx yz z y x G G G τ τ τ σ σ σ ν ν ν ν ν ν γ γ γ ε ε ε 1/ 1/ 1/ 1/ / / / 1/ / / / 1/ = xy y x xy y x xy y yx x xy y x G τ σ σ ν ν γ ε ε 1/ / / 0 / 1/
22 Assignment Textbook: Mechanical behavior of materials HW 5.11, 5.12, 5.37 Jiangyu Li,
Dr. D. Dinev, Department of Structural Mechanics, UACEG
Lecture 4 Material behavior: Constitutive equations Field of the game Print version Lecture on Theory of lasticity and Plasticity of Dr. D. Dinev, Department of Structural Mechanics, UACG 4.1 Contents
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