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1Variable Definition Notes & comments Pi-theorem (also definition of physical quantities,…)Physical similarity means that Physical similarityall Pi-parameters are equalGalileo-number (solid mechanics)Reynolds number (fluid mechanics)Extended base dimension systemLectures 1-3 and PS2Important concepts include the extended base dimension system, distinction between units and dimensions, the formal Pi-theorem based procedure and the concept of physical similarity. Applications include calculation of physical processes like atomic explosion, drag force on buildings etc.2Variable Definition Notes & comments rvvr= dxr/dtVelocity vectorrraa = dvr/dtAcceleration vectorUnit vectors that er r r1,e2,e3define coordinate system = basisNormal vector rAlways points outwards of ndomain consideredForce vector (force that acts on a material point)Angular momentumrxxr= x er+ xr r1 1 2e2+ x3e3Position vectorrr rpp = mv = m(v1er1+ v2erv er2+3 3)Linear momentumxr rr r r ri× pixi× pi= xi× mivirFFr= F er r rx x+ Fyey+ FzezCovered in lecture 4 and PS1Basic definitions of linear momentum, angular momentum, normal vector of domain boundaries3Variable Definition Notes & comments Dynamic resultant theoremrvdefrChange of linear momentum d( p) / dt = d(mv)/ dt = Fis equal to sum of external forcesDynamic moment theoremChange of the angular motion of a discrete system of i = 1,N particles is equal to the sum of the moments (or torque) generated by external forcesNewton’s three lawsStatic EQ (solve truss problems) 1. Every body continues in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed upon it.2. The change of motion is proportional to the motive force impresses, and is made in the direction of the right line in which that force is impressed.3. To every action there is always opposed an equal reaction: or, the mutual action of two bodies upon each other are always equal, and directed to contrary parts.d∑N()r rdef∑Nxi×mivi=(rr)Nrx × Fext exti=∑Mdti ii=1 i=1 i=1Lecture 4: These laws and concepts form the basis of almost everything we’ll do in 1.050.The dynamic resultant theorem and dynamic moment theorem are important concepts that simplify for the static equilibrium. This can be used to solve truss problems, for instance.4Variable Definition Notes & comments REV=Representative volume element‘d’=differential elementMust be:(1) Greater than any in homogeneity (grains, molecules, atoms,..)(2) Much smaller than size of the systemAtomic bondsO(Angstrom=1E-10m)Grains, crystals,…REVdΩContinuum representative volume elementREV∂ΩNote the difference between Surface of domain Ω‘d’ and ' o∂'peratorLecture 5The definition of REV is an essential concept of continuum mechanics: Separation of scales, i.e., the three relevant scales are separated sufficiently. There are three relevant scales in the continuum model. Note: The beam model adds another scale to the continuum problem – therefore the beam is a four scale continuum model. Skyscraper photograph courtesy of jochemberends on Flickr.5Variable Definition Notes & comments ⎛T⎞rxr r⎜ ⎟Stress vector T(n, x) =⎜Ty⎟⎜ ⎟(note: normal always points⎝Tz out ⎠of domain)Force density that acts on a material plane with normal nrat pointxrTr(nr, xr) =σ(xr)⋅nrStress matrixStress tensorpPressure (normal force per area that compresses a medium)σ=σerij i⊗ erjLecture 5, 6, 7These concepts are very important. We started with the definition of the stress vector that describes the force density on a particular surface cut. The stress tensor (introduced by assembling the stress matrix) provides the stress vector for an arbitrary plane (characterized by the normal vector). This requirement represents the definition of the stress tensor; by associating each entry with two vectors (this is a characteristic of a second order tensor). The pressure is a scalar quantity; for a liquid the pressure and stress tensor are linked by a simple equation (see next slide).6Variable Definition Notes & comments Differential equilibrium (solved by integration)onS :don∂Ω : T = T(n)EQ for liquid (no shear stress=material law)Divergence theorem (turn surface integral into a volume integral)divσ+ρ(gr− ar) = 0Differential E.Q. written out for cartesian C.S.In cartesian C.S.Lecture 5, 6, 7We expressed the dynamic resultant theorem for an arbitrary domain and transformed the resulting expression into a pure volume integral by applying the divergence theorem. This led to the differential EQ expression; each REV must satisfy this expression. The integration of this partial differential equation provides us with the solution of the stress tensor as a function of all spatial coordinates.7Variable Definition Notes & comments Divergence of stress tensor in cylindrical C.S.Divergence of stress tensor in spherical C.S.PS 4 (cylindrical C.S.)This slide quickly summarizes the differential EQ expressions for different coordinate systems.8Variable Definition Notes & comments Section quantities - forces==σStress tensor beam geometryBeam geometrySection quantities - momentsNxzyh,b << lzSectionLecture 8Introduction of the beam geometry. The beam is a ‘special case’ of the continuum theory. It introduces another scale: the beam section size (b,l) which are much smaller than the overall beam dimensions, but much larger than the size of the REV.9Variable Definition Notes & comments Beam EQ equations+BCs+BCs2D planar beam EQ equationszxLectures 8, 9The beam EQ conditions enable us to solve for the distribution of moments and normal/shear forces. The equations are simplified for a 2D beam geometry.10Variable Definition Notes & comments EQ for truss structures (S.A.)Strength criterion for truss structures (S.C.)σ0Tensile strength limitFσmax0=A0PPFmax= FbondNA0# bonds per area A0Strength per bondA0Concept: Visualization of the ‘strength’Number of atomic bonds per area constant due to fixed lattice parameter of crystal cellTherefore finite force per area that can be sustainedxxx: marks bonds that break at max forceFbondLecture 10, PS 5 (strength calculation)11Variable Definition Notes & comments Mohr plane (τ and σ)Mohr circle(Significance: Display 3D stress tensor in 2D)Tr(xr,nr) =σnr+τtrσ,τBasis in Mohr planePrincipal stressesPrincipal stress directionsPrincipal stresses and directions obtained through eigenvector


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