equations in materials science

This set of common equations used in Materials Science and Engineering supports the open educational resources (OERs) available through the CORE-Materials repository.

This document has been released under a Creative Commons Attribution 2.0 licence courtesy of CORE Materials 0

www.core.materials.ac.uk

Set of common equations

used in Materials Science and Engineering

Young’s modulus

Hardness test

Electrical testing

Heat treatment

Surface treatment

Composites

Polymers

Diffusion

Phase transformations

Recovery & recrystallization

Deformation

Corrosion

Resistivity / Conductivity

Lasers

http://www.materials.ac.uk/index.as

Young’s modulus

E=σε

E = Young's modulus (modulus of elasticity) (Pa)σ = stress (Pa)ε = strain

Young’s modulus is the relationship between stress and strain where a material is being deformed elastically and the stress/strain curve is linear.

TeX format: E = \frac{\sigma}{\varepsilon}

True stressσ t=σ (1+ε )

σt = true stress (Pa)σ = engineering stress (Pa)ε = engineering strain

Engineering stress is the form commonly used in applications, testing and stress-strain curves but does not take into account the reduction in cross-sectional area as a specimen plastically deforms. The true stress takes this factor into account.

TeX format: \sigma_{t} = \sigma{(1 + \varepsilon)}

True strainε t=ln (1+ε)

εt = true strainε = engineering strain


UK Centre for Materials Education

2010, The University of Liverpool

Engineering strain is the form commonly used in quantising, testing and stress-strain curves but does not take into account the reduction is cross-sectional area as a specimen plastically deforms. This is the true strain.

TeX format: \varepsilon_{t} = ln{(1 + \varepsilon)}

Hoop stress (thin wall assumption)

σ h=Prt

σh = Hoop stress (Pa)P = internal pressure (Pa)r = cylinder radius (m)t = cylinder wall thickness (m)

Gives the value of stress at the inner radius of a cylindrical shaped container using a thin assumption that the radius is at least ten times the wall thickness.

TeX format: \sigma_h = \frac{Pr}{t}

Hooke’s lawF=−kx

F = reaction force (N)k = spring constant (kgs-2)x = displacement (m)

Hooke’s Law describes the repulsive force exerted by an elastically deformed material (or spring). This equation is analogous to Young’s modulus.

TeX format: {F} = {-kx}

Poisson’s ratio


ν=−εtransεaxial

=−ε xε y

ν = Poisson's ratioεtrans, εx = transverse strainεaxial, εy = axial strain (load direction)

Poisson’s ratio is a material property strain in the transverse direction to the strain in the axial (applied load) direction. Applies to elastic deformation only.

TeX format: \nu = -{\frac{\varepsilon_{trans} }{\varepsilon_{axial} }} = - {\frac{\varepsilon_{x} }{\

varepsilon_{y}}}

Steady-state creep equationdεdt

=Aσ ne(−QRT )

dε/dt = creep (strain) rate (s-1)A = constantσ = constant applied stress (Pa)n = stress exponentQ = activation energy (J)T = temperature (K)R = gas constant (J.kg-1K-1)

Creep is the time-dependent deformation when subjected to a constant load beneath the yield strength. This is the equation for steady state creep (aka secondary creep).

TeX format: \dot{\varepsilon} = A\sigma^ne^{(\frac{-Q}{RT})}

Crack strain energy release rate


G=π ( σ 02cE )G = strain energy release rate (Jm-1)σ0 = fracture toughness (Pa)c = crack length (m)E = Young’s modulus (Pa)

During crack growth in a material, strain energy release rate was defined by A. A. Griffith in the following equation. The Griffith Criterion states that for crack growth to occur, this value must be greater than rate of energy absorption.

TeX format: G = \pi\left(\frac{{\sigma_0}^2c}{E}\right)

Vickers hardness test

HV=2sin (68 ° ) F

d2

HV = Vickers Hardness number (Pa)F = indenter load (N)d = mean indent diameter (m)

The Vickers hardness test is a measure of a material’s resistance to permanent deformation at the surface using a diamond indenter. Although not strictly speaking a property, hardness is often used in materials selection process. The equation relates specifically to the Vickers testing process and equipment.

TeX format: HV = \frac{2sin(68\degree )F}{d^2}

Brinell hardness test

BHN= Fπ2D ¿¿

BHN {BHN} = Brinell Hardness Number (Pa)D = indenter ball diameter (m)d = mean indent diameter (m)


The Brinell hardness test is a measure of a material’s resistance to permanent deformation at the surface using a spherical indenter (ball bearing). Although not strictly speaking a property, hardness is often used in materials selection process. The equation relates specifically to the spherical testing process and equipment.

TeX format: BHN = \frac{F}{{\pi\over 2}D(D - \sqrt{D^2-d^2})}

Knoop hardness test

HK= P

Cp L2

HK {HK} = Knoop hardness (Pa)P = indenter load (N)Cp = indenter shape correction factorL = largest indent length/diameter (m)

The Knoop microhardness test is a measure of a material’s resistance to permanent deformation at the surface using a pyramid diamond indenter. Although not strictly speaking a property, hardness is often used in materials selection process. The equation relates specifically to the Knoop testing process and equipment, used primarily for brittle and thin materials at a small scale.

TeX format: HK = {P \over {C_pL^2}}

Coffin-Manson relation∆ε p2

=εf' (2N )c

Δεp/2 = plastic strain amplitudeε’f = fatigue ductility coefficient2N = number of reversals to failurec = fatigue ductility exponent - empirical constant


The Coffin-Manson relation is a low-cycle fatigue strain classification relating fatigue strain size to the number of cycles.

TeX format: {\Delta \varepsilon_{p}\over2} = {\varepsilon^{'}_f(2N)^c}

Miner’s rule

∑i=1

k niN i

=C

k = different stress magnitudes in a spectrumni = number of cyclesNi = number of cycles to failureC = experimental constant between 0.7 and 2.2, usually for design purposes, C is assumed to be 1

Miner’s rule is used to estimate fatigue life left of a sample which has undergone cyclic stresses at varying amplitudes. Typically used in high-cycle applications.

TeX format: {\sum_{i=1}^{k}{n_i\over{N_i}} = C}

Paris-Erdogan lawdadN

=C ∆ Km

da = the crack length (m) dN = number of load cyclesC and m = material constantsΔK = range of stress intensity factor (m)

The Paris-Erdogan law states the relationship of the stress intensity factor range to sub-critical crack growth under a fatigue stress regime.


TeX format: {da\over{dN}}= {C\Delta K^{m}}

Hall-Petch relationship

σ y=σ 0+k y√d

σy = yield stress (Pa)σ0 = material constant (Pa)ky = strengthening coefficient (Pa.m0.5)d = grain diameter(m)

The Hall-Petch relationship illustrates the relationship between yield strength and grain size. Only holds down to ≈10nm grain size after which plastic deformation uses a different mechanism and yield strength decreases.

TeX format: \sigma_y =\sigma_0 + {k_y \over \sqrt{d}}

Fick’s first law of diffusion

J=−D ∂ϕ∂ x

J = diffusion flux (mol.m-2s-1)D = diffusivity (m2s-1)Φ = concentration (mol.m-3)x = position (m)

Fick’s first law of diffusion states that flux moves from regions of high concentration to regions of low concentration with a magnitude that is proportional to the concentration gradient.

TeX format: J = -D\frac{\partial \phi }{\partial x}


Fick’s second law of diffusion∂ϕ∂t

=D ∂2ϕ∂ x2

D = diffusivity (m2s-1)Φ = concentration (mol.m-3)x = position (m)t = time (s)

Fick’s second law of diffusion characterises how the concentration field at a point will changes with time.

TeX format: {\partial\phi\over\partial t} = D{\partial^2\phi\over\partial x^2}

Darken’s equations

J A' =−~D( ∂CA

∂ x )∂CA

∂t=~D ( ∂C A

∂x )JA’ = diffusion flux (mol.m-2s-1)D = interdiffusion coefficient (m2s-1)CA = concentration (mol.m-3)x = position (m)t = time (s)

The Darken regime is an adaptation of Fick’s Laws for substitutional diffusion by the motion of vacancies. The Darken equation is an adaptation of Fick’s laws for this purpose.

TeX format: 1. {J_A}' = - \tilde{D}\left (\frac{\partial C_A}{\partial x}\right)


2. \frac{\partial C_A}{\partial t} = \tilde{D} \left(\frac{\partial C_A}{\partial x}\right)

Clausius-Clapeyron relationdPdT

= LT ∆V

dP/dT = slope of the coexistence curve (JK-1m-3)L = latent heat (J)T = temperature (K)ΔV = volume change of the phase transition (m3)

The Clausius-Clapeyron relation represents the mathematical form of the coexistence curve (aka lines of equilibrium) on a Phase diagram. The coexistence curve marks the points on a pressure-temperature plot where multiple phases can coexist.

TeX format: {dP\over dT} = \frac{L}{T\Delta V}

Avrami equationy=1−exp−k tn

y = fraction of material changedk = material dependant constant (no units)n = material dependant constant (no units)t = time (s)

The Avrami equation describes the change of matter from one state to another at a constant temperature. A graph of y against log(t) is characterised by an initially slow rate of change (corresponding to nucleation of the new phase) followed by a faster, steady rate (growth of the new phase), and finally a decrease in rate (as the driving force behind the phase change reduces).

TeX format: y = 1 - \exp^{- kt^{n}}


Scheil equationsCL=C0 ( f L)k−1

CS=k C0 (1−f S )k−1

CL = Concentration of solute in the liquid phase (mol.m-3)CS = Concentration of solute in the solid phase (mol.m-3)C0 = Concentration of solute in the liquid phase at start of solidification (mol.m-3)f L = Fraction of total mass in the liquid phase (no units)f S = Fraction of total mass in the solid phase (f L+ f S=1) (no units)k = Distribution coefficient of solute (k=CS/CL) (no units)

The Scheil equations describe the concentration of a solute in the solid and liquid phases during solidification. This equation is commonly used in metallurgy, during the solidification of alloys and in semiconductor wafer manufacture for the solidification of single crystals of semiconductor materials.

TeX format: C_L=C_0(f_L)^{k-1}

C_S=kC_0(1-f_S)^{k-1}

Black’s equation

MTTF=Aw j−n exp( QkT )

MTTF = Mean time to failure (s)A = material/geometry dependant constantw = width of metal wire, sometimes included with the constant, A (m)j = current density (A.m-2)n = material/geometry dependant constant (typically = 2, no units)Q = activation energy for electromigration (eV)k = Boltzmann’s constant (8.617 eV.K-1)T = Temperature (K)In integrated circuit manufacture the Mean Time To Failure (MTTF) equation is used to predict the lifetime of nano-scale connections between components such as transistors. By running short experiments at elevated temperatures the constants in the equation can be calculated, allowing the time to failure (defined as the failure of 50% of interconnect lines) at normal operating temperatures to be predicted.


TeX format: MTTF = Awj^{-n}exp^{\frac{Q}{kT}}

RC (interconnect) delay

RCdelay≅ ρε L2

w2

RCdelay = (RC: Resistance & Capacitance) Time taken for a signal to pass through an interconnect wire (s)ρ = Resistivity of interconnect wire (Ωm)ε = Permittivity of dielectric material surrounding wire (F.m-1)L = Length of metal interconnect (m)w = Width or separation of interconnects (m)As integrated circuits decrease in size and more transistors are squeezed into the same amount of space, the interconnecting wires remain at roughly the same overall length, whilst the width of dielectric layers between them is reduced. This introduces a capacitance and resistance which must be overcome for a signal to pass.

TeX format: RC delay \approx \rho \varepsilon \frac{L^2}{w^2}

Equivalent oxide thickness (EOT)

EOT=t high−k ( k SiO2k high−k )EOT = Equivalent oxide thickness (m)t high−k = thickness of the high-k dielectric layer (m)k SiO 2

= relative permittivity of SiO2 (3.9 F.m-1)k high−k = relative permittivity of SiO2 replacement (F.m-1)As integrated circuits are miniaturised, quantum tunnelling of electrons between adjacent interconnects and transistors becomes a problem, causing signal loss and the generation of heat. To combat this, replacement dielectrics can be used, for example HfO2, which has a k ≅ 21 F.m-1. This allows a thicker dielectric layer to be used to give the same performance as SiO2 with a reduced likelihood of quantum tunnelling.

TeX format: EOT = t_{high-k} \left(\frac{k_{SiO_2}}{k_{high-k}}\right)


Critical thickness for strained oxide layer

hc=2s.1−νE εm

2.P

hc = Critical oxide thickness (m)ν = Poisson’s ratio for strained material (no units)s = Mean separation of misfit dislocations (m)E = Young’s modulus of strained material (Pa)εm = Misfit strain (no units)P = Dislocation energy per unit length (N.m-1)One method of increasing the speed of a microprocessor is to deposit strained channel material between gates in MOSFETs. This effectively reduces the delay in and between transistors. Electrons passing from one gate to another encounter less material and as such undergo less scattering, increasing their apparent velocity. The thickness of the strained material is limited by the strain field associated with the lattice mismatch between the channel and substrate material, and at a critical thickness will introduce dislocations into the deposited material, which are detrimental to the effect sought.

TeX format: h_c = \frac{2}{s}\cdot\frac{1-\nu}{E{\varepsilon_m}^2}\cdot P

Resistivity / Conductivity

ρ=1σ=R A

L= 1N e e μe

ρ = resistivity (Ωm)σ = conductivity (Sm-1)R = resistance (Ω)A = cross-sectional area (m2)L = length of sample (m)Ne = density of free electrons (m-3)e = electron charge (C)μe = electron mobility (m2V-1s-1)

Material property describing how well a material can conduct electricity. Materials with high resistivity (low conductivity) are known as insulators. Materials with low resistivity are conductors. This is generally a good indication of a material’s ability to conduct heat. Can be effected by environmental (thermodynamic) factors.


TeX format: \rho = {1\over\sigma} = R{A\over L}

Piezoresistivity

ρσ=

∂ ρρ∂LL

ρσ = piezoresistivity∂ρ = change in resistivity∂L = change in lengthρ = original resistivityL = original length

Stress over a conductor will induce a strain and therefore change the resistance of the conductor. In semiconductors, strain also causes a change in the material’s resistivity, causing a much larger change in resistance. This change in material conductivity brought about by strain is known as piezoresistivity. Not related to the piezoelectric effect. Used primarily in strain gauges.

TeX format: \rho_\sigma = \frac{\left(\frac{\partial\rho}{\rho}\right)}{\left(\frac{\partial L}{L}\right)}

Capacitance

C=QV

C = capacitance (F)Q = charge (C)V = potential difference (V)

The electrical capacitance of a material or, more usually, a device is defined as the amount of charge than can be stored for a given electric potential (driving force).

TeX format: C = \frac{Q}{V}

Capacitance (parallel plate)


C=ε0 εr EA

V

C {C} = capacitance (F)ε0 = permittivity of free space (Fm-1)εr = relative permittivityE = electric field strength (Vm-1)

The most basic type of capacitor involves running a potential difference over two parallel plates separated by a dielectric, causing charge to build up on the plates. This is the defining equation of this arrangement, which underpins every type of capacitor.

TeX format: C = \frac{\varepsilon_0\varepsilon_r EA}{V}

Degree of Crystallinity

%C=ρcρ s (

ρs− ρaρc−ρa )×100

%C = crystallinity of sample (%)ρc = density of crystalline solid (kg.m-3)ρa = density of amorphous solid (kg.m-3)ρs = density of sample with unknown % crystallinity (kg.m-3)

This equation is used to determine the percentage crystallinity for a semi-crystalline sample by the measured density of the sample and of the densities of the fully crystalline and fully amorphous states (assuming no voids are present).

TeX format: \% C = \frac{\rho_c}{\rho_s} \left ( \frac{\rho_s – \rho_a}{\rho_c – \rho_a} \right ) \times 100


Degree of polymerisation

n=M℘

m

n = Degree of polymerisation, also = average number of mers per polymer chain (no units)M℘ = Molecular weight of polymer (g.mol-1)m= Molecular weight of mer (g.mol-1)

The degree of polymerisation gives the average number of mer units joined together to form each polymer chain. This allows differentiation between polymers with the same composition but different chain lengths, which can have significant effects on the overall material properties.

TeX format: n = \frac{M_{wp}}{\bar{m}}


Interlaminate Shear Strength (ILSS)

ILSS=3Pmax

4 Bd

ILSS = Interlaminate shear strength (Pa)Pmax = Maximum force reached during test (N)B = Sample width (m) d = Sample thickness (m)

This equation is used to determine the interlaminate shear strength of a composite material, based upon the force at failure during a 3 point bend test. The test is typically standardised, with B = 10mm, d = 2mm and the distance between the two support points = 20mm.

TeX format: ILSS = \frac{3 P_{max}}{4Bd}

Beam theory method for DCB test

G Ic=3 Pδ2Ba

G Ic = Mode I fracture energy (J.m-2)P = Force (N)δ = Displacement at point of force (m)B = Sample width (m)a = Distance from P to crack tip (m)The Double Cantilever Beam (DCB) test is used to assess the mode I failure strength of a composite sample. This equation gives a rough value for the mode I fracture energy, as it assumes that the beam is perfectly built in at the crack tip.

TeX format: G_{Ic} = \frac{3 P \delta}{2Ba}


Corrected beam theory method for DCB test

G Ic=3 Pδ

2 B (a+∆ )

G Ic = Mode I fracture energy (J.m-2)P = Force (N)δ = Displacement at point of force (m)B = Sample width (m)a = Distance from P to crack tip (m)∆ = Distance ahead of crack tip to area where beam is perfectly built in (m)This modified beam theory equation assumes that the crack is effectively longer than that measured during the Double Cantilever Beam (DCB) test. The additional crack length is calculated from the sample’s compliance (C=δ /P). Plotting C1/3 against a yields a linear graph in which the line of best fit crosses the x-axis at some negative value of a. The magnitude of this value is the additional crack length, ∆.

TeX format: G_{Ic} = \frac{3 P \delta}{2B(a+\Delta)}

Experimental compliance method for DCB test

G Ic=nPδ2 Ba

G Ic = Mode I fracture energy (J.m-2)n = gradient of log (C) vs log (a) (no units)P = Force (N)δ = Displacement at point of force (m)B = Sample width (m)a = Distance from P to crack tip (m)In the DCB test, a measure of mode I fracture energy can be obtained using the experimental compliance method. It is assumed that the compliance (C=δ/P) is related to the crack length (a) by the equation

C=k an, where k and n are constants. A plot of log (C) vs log (a) gives a linear graph with the gradient equal to n.

TeX format: G_{Ic} = \frac{n P \delta}{2Ba}


Mode II fracture energy for ENF test

G IIc=9a2 P2

16 E B2h3

G IIc = Mode II fracture energy (J.m-2)a = Distance from end support to crack tip (m)P = Force (N)E = Young’s modulus (Pa)B = Sample width (m)2h = Sample thickness (m)The End Notch Flexure (ENF) test is used to determine the mode II fracture energy of a composite material.

TeX format: G_{IIc} = \frac{9a^2P^2}{16EB^2h^3}

Mixed mode (I/II) fracture energy for SCB test

G I / IIc=3ma2P2

2B

G I / IIc = Mixed mode (I/II) fracture energy (J.m-2)m = gradient of line from plot of C vs a3 (N-1m-2)a = Distance from end support to crack tip (m)P = Force (N)B = Sample width (m)For the Single Cantilever Beam (SCB) test, the specimen compliance (C=δ /P) is assumed to follow the

equation C=C0+ma3. Plotting C against a3 yields a linear graph with a gradient equal to m.

TeX format: G_{I/IIc} = \frac{3ma^2P^2}{2B}


Fourier’s law

J= q̇A

=−kT 2−T 1x

J = flux (Jm-2s-1)dq/dt = heat transfer rate (Js-1)A = cross-sectional area (m-2)T2 = hot source (>T1) temperature (K)T1 = cold source (<T2) temperature (K)x = distance between sources. (m)k = thermal conductivity of substance between sources. (Js-1m-1K-1)

Fourier’s Law of thermal conduction can have several forms and describes the transfer of thermal energy from a hot source to a cold one.

TeX format: J = \frac{\dot{q}}{A} = -k\frac{T_2 - T_1}{x}

Arrhenius equation

k=Ae−E a

RT

k = reaction rate coefficient (s-1)A = prefactor (mol-1)T = temperature (K)R = universal gas constant (JK-1)Ea = Activation energy (J.mol-1)

The Arrhenius equation quantises the effect of temperature on a chemical reaction rates.

TeX format: {k}= {Ae^{ -E_a\mathrm/RT}}


Dalton’s law of partial pressures

Ptotal=∑i=1

n

p i

Ptotal = total pressure (Pa)p1, p2 .. = partial pressures (Pa)

Dalton’s law states that the pressure exerted by a gaseous mixture is equal to the sum of the partial pressures of the constituent gasses. Remains valid up to high pressures.

TeX format: P_{total} = \sum_{i = 1}^{n}p_i

Nernst equation

Ee=E0−2.303RT

zFlog| [reduced ]

[oxidised ]|Ee = electrode potential (V)E0 = standard potential (V)R = universal gas constant (J K-1 mol-1)T = temperature (K)z = number of moles of electronsF = Faraday constant (Cmol-1)

The Nernst equation links the equilibrium potential of an electrode to its standard potential and the concentrations or pressures of the reacting components at a given temperature. It describes the electrode potential for a given reaction as a function of the concentrations (or pressures) of all participating chemical species.

The notation [reduced] represents the product of the concentrations (or pressures where gases are involved) of all the species that appear on the reduced side of the electrode reaction, raised to the power of their stoichiometric coefficients. The notation [oxidised] represents the same for the oxidised side of the electrode reaction.

TeX format: E_e = E^0 - \frac{2.303RT}{zF}\log{ \left|\frac{[reduced]}{[oxidised]}\right |}

Butler-Volmer equation


I=i0(exp (αnFηRT )−exp(−(1−α )nFηRT ))

I = current (A)i0 = exchange current density (A)α = transfer coefficient F = Faraday constant (Cmol-1)R = universal gas constant (J K-1 mol-1)T = temperature (K)n = number of electronsη = overpotential - the difference between electrode potential and equilibrium potential (V)

The Butler-Volmer equation describes how electrical current in an electrochemical reaction depends on the electrode potential, considering both a cathodic and an anodic reaction occur on the same electrode.

TeX format: I = i_0\left( exp\left( \frac{\alpha nF\eta}{RT}\right) - exp\left(\frac{-(1-\alpha)nF\eta}{RT}\

right)\right)

Tafel equation

ΔV=η=A ln ( ii0 )ΔV = overpotential (V)A = Tafel slope (V)i {i} = current density (Am-2) {Am^{-2}}i0 {i_0} = exchange current density (Am-2) {Am^{-2}}

The Tafel equation relates the electrochemical reaction rate to overpotential.

TeX format: \Delta V = A ln\left(\frac{i}{i_0}\right)

Energy density of a laser beam


Eρ=2PπrV

Eρ = Laser energy density (J.m-2)P = Laser power (W)r = Beam radius (m)V = Scan velocity (m.s-1)

In materials processing, the above equation describes the energy density of a laser beam (also known as beam exposure). At high enough energy densities lasers can be used to cut through sheet metals, weld materials together, sinter powders together to form solids, etc.

TeX format: E_{\rho} = \frac{2P}{\pi r V}

Required laser power in SLM

P=2 rVtρCT m

1−R

P = Laser power required to cause melting (J.m-2)r = Laser beam radius (m)V = Scan velocity of beam (m.s-1)t = Layer thickness (m)ρ = Powder density (kg.m-3)C = Specific heat capacity of powder (J.kg-1.K-1)T m = Melt temperature of powder (K)R = Reflectivity of powder (no units)The rapid prototyping methods Selective Laser Melting (SLM) and Selective Laser Sintering use a medium to high power laser beam to fuse adjacent powder particles together. By scanning the beam back and forth across a powder bed, a thin layer of sintered or melted material is formed. The powder bed then drops by one layer thickness, fresh powder is wiped across the surface and the process is repeated. In this way a complete 3D solid metallic model can be produced.

TeX format: P = \frac{2rVt\rho C T_m}{1-R}

Minimum spot size for a laser


W 0=4M 2 fλπ DL

W 0 = Minimum spot radius (m)M = Beam mode (no units)f = Focal length of lens (m)λ = Wavelength of laser (m)DL = Diameter of laser beam at lens (m)This equation gives the minimum spot radius for a laser beam, based on beam mode and optic geometry. This corresponds to the part of the beam with highest radiant power.

TeX format: W_0 = \frac{4M^2f \lambda}{\pi D_L}

Irradiance at minimum spot in a laser beam

H ( r ,0 )=Hmaxexp−( 2r

2

W 02 )

H ( r ,0 ) = Irradiance at a distance r from beam centre at z = 0. (W.m-2)Hmax = Irradiance at centre of beam at z = 0. (W.m-2)r = radius from centre of beam (m)W 0 = Minimum spot radius (m)In rapid prototyping processes such as Stereolithography (SLA), it is important to determine the laser beam’s interaction with the prototype resin. Specifically, the distribution of the laser beam’s radiant power at the surface of the resin and below will determine the voxel (volume pixel) geometry. The above equation describes the distribution of beam power at zero depth when the laser beam’s minimum radius coincides with the resin surface.

TeX format: H_{(r,0)} = H_{max} exp^{-\left(\frac{2r^2}{{W_0}^2}\right)}

Irradiance distribution within SLA resin


H ( r , z)=H (r , 0) exp(− zDp )

H ( r , z) = Irradiance at a distance r from beam centre at a depth of z (W.m-2)H ( r ,0 ) = Irradiance at a distance r from beam centre at z = 0. (W.m-2)z = Depth below resin surface (m)D p = Beam penetration depth, (when irradiance has reached 37% that of the surface) (m)In rapid prototyping processes such as Stereolithography (SLA), it is important to determine the laser beam’s interaction with the prototype resin. Specifically, the distribution of the laser beam’s radiant power at the surface of the resin and below will determine the voxel (volume pixel) geometry. The above equation describes the distribution of beam power at a depth = z when the laser beam’s minimum radius coincides with the resin surface, and assumes a Beer-Lambert law of absorption for the resin.

TeX format: H_{(r,z)} = H_{(r,0)} exp^{\left(\frac{-z}{D_p}\right)}

Laser exposure within resin for SLA

E( r , z )=√ 2π [ PLW 0V s

] .exp−( zDp−2 r 2

W 02 )

E( r , z ) = Laser exposure at depth z and distance r from the beam centre (J.m-2)P = Beam power (W)W 0 = Effective beam radius (m)V s = Scan velocity of beam (m.s-1)z = Depth below resin surface (m)D p = Beam penetration depth, (when irradiance has reached 37% that of the surface) (m)r = radius from centre of beam (m)In the rapid prototyping process stereolithography (SLA), the resin cures when exposed to energies above a critical exposure. The above equation describes the exposure on the surface of and within the resin for a laser beam of power PL moving at a velocity V s.

TeX format: E_{(r,z)} = \sqrt{\frac{2}{\pi}} \left [\frac{P_L}{W_0 V_s} \right ] \cdot exp^{-\left (\frac{z}{D_p}-\frac{2r^2}{{W_0}^2} \right )}


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