elastic deformation
合集下载
- 1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
- 2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
- 3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。
Response to Stress Elastic: reversible strain (e.g a spring) Plastic: permanent strain (e.g. a bent paperclip) Fracture: propagation of a crack (e.g. breaking glass) Viscous: flow of a liquid
For now we are concerned with the initial reversible, usually linear, part of the curve that precedes yield and permanent deformation.
Elastic modulus in tension
Bulk P=K.DV/V Poisson’s ratio, n = Dw/DL ~1/2
And many more.
Elastic Bending In bending, one surface is in tension, one is in compression and the center is neutral. Shear stress is highest at the center.
Stiffness is bad for:
Trees Buildings in earthquakes cords Structures subject to impacts: bicycle frames, auto suspensions, bodies
Other moduli
Shear t=g. g~1/3E
Stiffness is good for:
Stiff: low elastic deflection under load Aircraft: stiff & light: aluminum alloy, carbon fiber composite Columns prone to elastic compressive buckling: pillars, legs, bicycle frames Radio masts
Visco-elastic: slow elastic strain and recovery (e.g. toffee)
Fatigue: slowly growing crack in cyclic loading
Creep: slow permanent strain due to vacancy motion (e.g. old lead pipe)
K = column effective length factor, whose value depends upon the conditions of end support of the column, as follows. For both ends pinned (hinged, free to rotate), K = 1.0. For both ends fixed, K = 0.50. For one end fixed and the other end pinned, K = 0.70. For one end fixed and the other end free to move laterally, K = 2.0. l = unsupported length of column, I= Moment of inertia
For a load P, length L, width b, thickness d, the stress at the top surface is:
For our weight lifter: P = 3000 Newtons, L =1 meter (2x bone length) b = 2 cm, d=2cm Stress = 562 MPa, enough to break the bone easily.
Linear elasticity Hooke’s Law, 1660 Deformation Load
Young’s modulus 1807 Stress = Modulus x Strain, = E .
Engineering Stress = Load/Original Area Engineering Strain = Extension/Length At large deformations the cross-sectional area reduces significantly from the original and we need to redefine stress. True Stress = Load/ Actual Area True strain = log (L/Lo)
Force = 98 N Area = πr2 = 78.5x10-6 m2 Stress = 1.2x106 Pa, 1.2 MegaPascals This will cause a strain (change in length/original length) in rod.
Mechanical Properties
In tension or bending elasticity may lead to deformation that is inconvenient. Buckling is also an important failure mode. A column under a centric axial load can exhibit elastic buckling, where it suddenly bends out and then kinks.
Mechanical testing needs a machine much stiffer than the sample. A load cell in the crosshead measures force as the sample is stretched at a constant rate by the screw drive (or compressed or bent)
Force, stress, strain and elastic deformation
Force, stress and strain Mass in kilograms, kg often called weight Produces, in gravity, Force in Newtons. 1kg produces a force of 9.8 Newtons A force pulling (or pushing) on a rod of a cross-sectional area A Produces a stress, measured in Pascals (Newtons per square meter) Stress = Force/(Area of cross section) For a weight of 10 kg hanging from a rod that is 1cm (0.01meters) in diameter
For now we are concerned with the initial reversible, usually linear, part of the curve that precedes yield and permanent deformation.
Elastic modulus in tension
Bulk P=K.DV/V Poisson’s ratio, n = Dw/DL ~1/2
And many more.
Elastic Bending In bending, one surface is in tension, one is in compression and the center is neutral. Shear stress is highest at the center.
Stiffness is bad for:
Trees Buildings in earthquakes cords Structures subject to impacts: bicycle frames, auto suspensions, bodies
Other moduli
Shear t=g. g~1/3E
Stiffness is good for:
Stiff: low elastic deflection under load Aircraft: stiff & light: aluminum alloy, carbon fiber composite Columns prone to elastic compressive buckling: pillars, legs, bicycle frames Radio masts
Visco-elastic: slow elastic strain and recovery (e.g. toffee)
Fatigue: slowly growing crack in cyclic loading
Creep: slow permanent strain due to vacancy motion (e.g. old lead pipe)
K = column effective length factor, whose value depends upon the conditions of end support of the column, as follows. For both ends pinned (hinged, free to rotate), K = 1.0. For both ends fixed, K = 0.50. For one end fixed and the other end pinned, K = 0.70. For one end fixed and the other end free to move laterally, K = 2.0. l = unsupported length of column, I= Moment of inertia
For a load P, length L, width b, thickness d, the stress at the top surface is:
For our weight lifter: P = 3000 Newtons, L =1 meter (2x bone length) b = 2 cm, d=2cm Stress = 562 MPa, enough to break the bone easily.
Linear elasticity Hooke’s Law, 1660 Deformation Load
Young’s modulus 1807 Stress = Modulus x Strain, = E .
Engineering Stress = Load/Original Area Engineering Strain = Extension/Length At large deformations the cross-sectional area reduces significantly from the original and we need to redefine stress. True Stress = Load/ Actual Area True strain = log (L/Lo)
Force = 98 N Area = πr2 = 78.5x10-6 m2 Stress = 1.2x106 Pa, 1.2 MegaPascals This will cause a strain (change in length/original length) in rod.
Mechanical Properties
In tension or bending elasticity may lead to deformation that is inconvenient. Buckling is also an important failure mode. A column under a centric axial load can exhibit elastic buckling, where it suddenly bends out and then kinks.
Mechanical testing needs a machine much stiffer than the sample. A load cell in the crosshead measures force as the sample is stretched at a constant rate by the screw drive (or compressed or bent)
Force, stress, strain and elastic deformation
Force, stress and strain Mass in kilograms, kg often called weight Produces, in gravity, Force in Newtons. 1kg produces a force of 9.8 Newtons A force pulling (or pushing) on a rod of a cross-sectional area A Produces a stress, measured in Pascals (Newtons per square meter) Stress = Force/(Area of cross section) For a weight of 10 kg hanging from a rod that is 1cm (0.01meters) in diameter