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MECH 4650 Definitions
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Gravity
Terms in this set (88)
Normal stress
engineering measure of how forces are transmitted through a body from a force applied perpendicular to the surface
Shear stress
engineering measure of how forces are transmitted through a body from a force applied parallel to the surface
Normal strain
engineering measure of the relative deformation per unit length of a specimen
Shear strain
engineering measure of the relative deformation in the change in angle
Yield strength
the strength at which a material enters the plastic region of deformation. Commonly uses a strain offset of 0.002
Ultimate strength
maximum stress that can be sustained without fracture
Fracture Strength
Property of a material, stress as which material fractures
Moment of Interitia
the capacity of a cross section to resist bending. Considered with respect to a reference axis and the center of mass or centroid of the cross section. how spread out a material is from the central axis
Polar Moment of Inertia
The capacity of a circular cross section to resist torsion. How spread out the material is from its rotational axis stiffness of geometry to resist torsion
Classical Beam Theory
planes perpendicular to the neutral axis after deofmration
moment
the tendency of a force to rotate an object when it acts at a perpendicular distance from the point considered, force applied at a distance.
Torque
a moment produced by some force acting at a perpendicular distance away from a point causing rotation of an object. A measure of force that is applied about an axis of rotation
Axial Load
forces are directed along the axis of the member, normal to the cross sectional area
Linear elastic
the behavior of a material to act linearly when a load is applied and resumes it original shape when the load is released as long as the stress is kept below the yield point
Isotropic
the mechanical properties of a material are independent of the direction considered, has same mechanical behaviors in all directions.
Homogeneous
a material in which properties are not a function of position within the material they are constant throughout
Youngs modulus
modulus of elasticity of a material, the slope of the linear elastic region on a stress strain graph
Coefficient of thermal expansion
a constant characteristic of a material that causes it to expand or contract due to a change in temperature.
Modulus of Rigidity
relates shear stress to strain
St Vernant's Principle
the difference between the effects of two different but statically equivalent loads become very small at sufficiently large distances from the load. Other than in the immediate vicinity of the points of application of loads, the distribution may be assumed independent of the actual mode of application of the load> We can replace a boundary condition with one that is energetically equivalent without affecting the state of stress away from the boundary condition
Superposition
states that the resultant stress or displacement at a point can be determined by first finding the stress or displacement caused by each component load separately on the member. The effect of combined loading on a structure can be obtained by determining the effects of the various loads separately and combining the results
Stress concentration factor
a ratio of the maximum stress to the applied tensile stress. refers to the stress concentration in loaded materials caused by holes, fillets, and flaws.
Stress Factor
ratio of the ultimate load. should be used in the design of a given structure upon a number of considerations
perfectly plastic
behavior of a material in which stress remains constant throughout the plastic region as strain increases
Stress redistribution
once material has yielded fully, any additional stress changes the stress distribution and is shifted to an area where it remains in equilibrium and forces are conserved
Plastic deofmation
deformation that is permanent or non-recoverable after the load is released
Poisson's Ratio
relates lateral and axial stress for elastic deformation, the negative ratio of a lateral and axial strains that results from an applied axial stress
Elastic/perfectly plastic
describes the behavious of a material when a load is applied, behaves linearly in the elastic region and stress is constant in the plastic region when strain increases
Shear Flow
a reorientation of the shear stress due to the presence of a traction free surface
Traction Free Surface
A surface with no normal shear stress component applied
Antielastic curgature
Curvature in transverse direction in a beam in bending due to poissons effect
Yield Criteria
Predict the onset of plastic deformation for a multi axial state of stress
Plastic Moment
Entire cross section has reached yield stress
Residual Stress
if a load is applied causing plastic deformation, there will be stress in the material after the load is removed
Stress Redesbtribution
in the plastic deformation, stress is linear until it reaches plastic deformation. The linear section increases to make up the flat portion where stress cannot go further.
Strain energy
the increase in energy associated with the deformation of the member
Strain energy density
of a material will be defined as the strain energy per unit volume; it will be seen that it is equal to the area under the stress-strain diagram of the material
Critical Buckling Load
aka euler buckling load. The maximum compressive load a member can support before failure due to buckling.
Effective Length
the length used to make the formula applicable to various end conditions
Radius of Gyration
used to calculate the slenderness ratio
Slenderness Ratio
l/r the length of the column over the radius of gyration is used in the calculation of the critical stress due to buckling
Castiglinaos Theorem
If the strain energy of a linearly elastic structure can be expressed as a function of generalized force Q, then the partial derivative of the strain energy with respect to generalised force gives the generalised displacement q in the direction of Q.
Statically Indeterminate
when statics equations are insufficient for determining the internal forces and reactions on the structure.
Plastic Torque
a fully plastic deformation, when the distribution of strain remains linear after the onset of yield in shear vs strain plot
Elastic Core
Zones where stress varies linearly with the strain
Von Mises Yield Criterion
a structure is safe as long as the max value of the distortion energy per unit volume in the material is smaller than the distortion energy per unit volume to cause yield in a tensile test specimen
Tresca
Maximum shear stress yield criteria (the difference between principal stresses must be less than the yield stress)
Mohr's Circle
a 2-D representation of the transformation law for the stress tensor in plane stress can be used when all non-zero stresses are in 1 plane and is on traction free surface
Principal Planes
planes that have no shear stress (only normal stresses or principal stresses) and are on oblique angles
Small Angle Approximation
simplification of basic trig functions which is approximately true when the limit where the angle approaches 0
Small Deformation Theory
the displacements of the material particles are assumed to be much smaller than any relevant dimension of the body and material properties can assumed to be unchanged by deformation.
Hydrostatic Stress
state of stress where only normal stresses exist and they are all equal
Plastic Deformation
any kind of deformation beyond yield stress of material results in permanent deformation
Neutral Axis
the axis of a cross section where there are no longitudinal stresses or strains.
Centroid
mean position of all the points in all the coordinate directions.
Eccentric loading
stress directed anywhere on a component other than where the component is designed to accept the force.
Plane Stress
the state of stress in which the normal stress and the shear stresses directed perpendicular to the x-y plane are assumed to be 0
Parallel Axis Theorem
used to determine the mass moment of inertia or a rigid body about any axis, given the body's moment of inertia about a parallel axis through the object's center of mass and the perpendicular distance between the axes.
Dummy Force, Moment
Fake load applied in the direction where deflection is to be determined using Castiglione's theorem.
Reciprocal work theorem
for linear elastic structure subjected to two forces P and Q the work done by P through the displacement produced by Q is equal to the work done by Q through the displacement produced by P
Degrees of Freedom
minimum number of independent coordinates that can specify the position of the system completely
Buckling
when structure fails under a compressive stress that is below the ultimate compressive stresses that the material is capable of handling
Anticlastic
having a curvature at a given point and in a particular direction, that is the opposite sign to the curvature at that point in a perpendicular direction
Fracture Toughness
The critical value of stress intensity
Creep
occurs when materials are subjected to high loads continuously, they may experience progressive elongation over time. it is a time dependent material response. since creep often takes a long time, so accelerated testing is often done to characterize a material's creep characteristics.
Endurance Limit
this is the point at below which failure to fatigue is unlikely. Note some materials such as aluminum don't exhibit a true endurance limit. The actual endurance limit is going to be influenced by many external factors called modifying factors.
SN Curve
a curve used to identify endurance limit and prevent failure due to fatigue. The stress by the number of cycles to failure.
Deviartoric (Distortional) Stress
Related to shape change, the stress left after subtracting the hydrostatic stress.
Volumetric (Hydrostatic) Stress
Related to volumetric change, simply the average between the 3 normal stress components.
Fatigue
Fatigue - a mode of failure experienced due to repeated loading or stress. The fatigue performance of materials is often characterized using an S-N curve. It is a time dependent material response.
Modifying Factors
The external factors that actually influence the actual endurance limit, such as corrosion, temperature, load type, material, size, surface finish, and more.
Mean Stress Effect
the mean stress between the maximum stress and minimum stress for a cyclic loading condition.
Goodman Type Analysis
Most often used analysis for machine components subjected to cyclic loads
Modified Goodman Type Analysis
Can be used to account for the effect of mean stress on fatigue performance
Variable Amplitude Loading
cyclic loading situation where the amplitude of the loading is variable and not constant.
Stress Signularity
The point in the material where the stress does not converge on one specific value, it goes till infinity.
Crack Length
the length of a crack. Failure occurs at the point when the crack length is at its critical point and the material fails.
Paris Law
...
Stress Intensity Factor
Characterize how quickly the stress goes to infinity by scaling the stress. The opening mode, shearing mode, tearing mode in which have these constants/factors for each mode, KI. This value should always be less than 1.
Se'
the pristine endurance limit. It is the stress below which failure due to fatigue is unlikely under ideal conditions.
Linear Damage Model
(# of periods)((n1 /nf1) + (n2/nf2) + (n3/nf3)) < 1 is safe, = 1 is a fail
Rheological Elements
there are 3 basic building elements: spring (elastic), stick-slip (plastic), damper (viscous)
Elastoplastic
Reacts as an elastic material until the yield point at which the material plastically deforms under the constant loadin
Viscoelastic
a property of a material that exhibits both viscous and elastic characteristics when undergoing deformation. It resists shear flow and strain linearly with time when a stress is applied. When stretched it will quickly return to its original state once the stress is removed. It exhibits time-dependent strain.
Relaxation
Material response where the stress rate decreases for a constant strain.
Modes of Fracture
Mode 1: Opening
Mode 2: Shearing of Sliding
Mode 3: Tearing
Dislocations
a displacement of part of the crystal lattice structure of a material.
Small Scale Yielding
is the argument for using the stress intensity factor and explains why it does (or sometimes doesn't) work.
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Verified questions
ENGINEERING
(a) Verify that $y_1 = x^3$ and $y_2 = |x|^3$ are linearly independent solutions of the differential equation $x^2 y'' - 4xy' + 6y = 0$ on the interval $(-\infty, \infty)$. (b) Show that $W(y_1, y_2 ) = 0$ for every real number $x$. (c) Verify that $Y_1 = x^3$ and $Y_2 = x^2$ are also linearly independent solutions of the differential equation in part (a) on the interval $(-\infty, \infty)$. (d) Find a solution of the differential equation satisfying $y(0) = 0, y' (0) = 0$. (e) Both linear combinations $y = c_1y_1 + c_2y_2$ and $Y = c_1Y_1 + c_2Y_2$ are solutions of the differential equation. Discuss whether one, both, or neither of the linear combinations is a general solution of the differential equation on the interval $(-\infty, \infty)$.
ENGINEERING
A hot dog can be considered to be a 12-cm-long cylinder whose diameter is 2 cm and whose properties are $\rho=980 \mathrm{kg} / \mathrm{m}^{3}, c_{p}=3.9 \mathrm{kJ} / \mathrm{kg} \cdot \mathrm{K}, k=0.76 \mathrm{W} / \mathrm{m} \cdot \mathrm{K},$ and $\alpha=2 \times 10^{-7} \mathrm{m}^{2} / \mathrm{s}.$ A hot dog initially at $5^{\circ} \mathrm{C}$ is dropped into boiling water at $100^{\circ} \mathrm{C}.$ The heat transfer coefficient at the surface of the hot dog is estimated to be $600 \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}.$ If the hot dog is considered cooked when its center temperature reaches $80^{\circ} \mathrm{C},$ determine how long it will take to cook it in the boiling water.
ENGINEERING
A plastic coating is applied to wood panels by first depositing molten polymer on a panel and then cooling the surface of the polymer by subjecting it to airflow at $25^{\circ} \mathrm{C}$. As first approximations, the heat of reaction associated with solidification of the polymer may be neglected and the polymer/wood interface may be assumed to be adiabatic. If the thickness of the coating is L=2 mm and it has an initial uniform temperature of $T_{i}=200^{\circ} \mathrm{C}$, how long will it take for the surface to achieve a safe-to-touch temperature of $42^{\circ} \mathrm{C}$ if the convection coefficient is $h=200 \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}$? What is the corresponding value of the interface temperature? The thermal conductivity and diffusivity of the plastic are $k=0.25 \mathrm{W} / \mathrm{m} \cdot \mathrm{K}$ and $\alpha=1.20 \times 10^{-7} \mathrm{m}^{2} / \mathrm{s}$, respectively.
ENGINEERING
A tank contains a mixture of 70% ethane and 30% nitrogen $(N_2)$ on a molar basis at 400 K, 200 atm. For 2130 kg of mixture, estimate the tank volume, in $m^3,$ using a. the ideal gas equation of state. b. Kay's rule together with data from the generalized compressibility chart. c. the ideal solution model together with data from the generalized compressibility chart.