List of symbols used in the Synthesizer Tool
Click here to see this page in full context.
  1. >
  2. Tools >
  3. Synthesizer Tool >
  4. Synthesizer Tool reference >
  5. List of symbols used in the Synthesizer Tool

List of symbols used in the Synthesizer Tool

A range of symbols are used in the equations describing the performance of the hybrid structures. These are defined, and their units given, in the tables below.

  1. Cellular structures
    1. Closed-cell foam model
    2. Open-cell foam model and triangulated lattice model
    3. Honeycomb model
  2. Composites (Simple Bounds) model
  3. Controlled Thermal Expansion model
  4. Cost
    1. Part cost estimator
  5. Multi-layer Materials model
  6. Sandwich Panels
    1. Balanced Sandwich Panel model

Cellular structures

Closed-cell foam model

Symbol Description Units (Metric)
relative density of cellular structure Relative density of cellular structure Dimensionless
ρs Density of solid material kg/m3
equivalent density Density of cellular structure kg/m3
φ Proportion of solid material in cell edges of a closed cell foam. So (1- φ) is the proportion in the cell faces. Dimensionless
Es Young's modulus of solid material GPa
equivalent Young's modulus Young's modulus of cellular structure GPa
σy,s , σts,s, σc,s , σflex,s Yield strength, tensile strength, compressive strength, and flexural strength of solid material MPa
K1c,s Fracture toughness of solid material (MPa).m1/2
l Average cell size m
a Maximum flaw size in solid material m
λs Thermal conductivity of solid material W/m.K
λg Thermal conductivity of the gas within the cells W/m.K
Cp,s Specific heat capacity of solid material J/kg.K
αs Thermal expansion coefficient of solid material mstrain/K
εr,s Dielectric constant of solid material Dimensionless
ρelec,s Electrical resistivity of solid material mW.cm
De,s Dielectric loss tangent of solid material Dimensionless

Top

Open-cell foam model and triangulated lattice model

Symbol Description Units (Metric)
relative density of cellular structure Relative density of cellular structure Dimensionless
ρs Density of solid material kg/m3
equivalent density Density of cellular structure kg/m3
Es Young's modulus of solid material GPa
equivalent Young's modulus Young's modulus of cellular structure GPa
σy,s , σts,s, σc,s , σflex,s Yield strength, tensile strength, compressive strength, and flexural strength of solid material MPa
K1c,s Fracture toughness of solid material (MPa).m1/2
l Average cell size m
a Maximum flaw size in solid material m
λs Thermal conductivity of solid material W/m.K
λg Thermal conductivity of the gas within the cells W/m.K
Cp,s Specific heat capacity of solid material J/kg.K
αs Thermal expansion coefficient of solid material mstrain/K
εr,s Dielectric constant of solid material Dimensionless
ρelec,s Electrical resistivity of solid material mW.cm
De,s Dielectric loss tangent of solid material Dimensionless

Top

Honeycomb model

Symbol Description Units (Metric)
relative density of cellular structure Relative density of cellular structure Dimensionless
ρs Density of solid material kg/m3
equivalent density Density of cellular structure kg/m3
Es Young's modulus of solid material GPa
equivalent Young's modulus Young's modulus of cellular structure GPa
Gs Shear modulus of solid material Dimensionless
σy,s , σts,s, σc,s , σflex,s Yield strength, tensile strength, compressive strength, and flexural strength of solid material MPa
K1c,s Fracture toughness of solid material (MPa).m1/2
l Average cell size m
a Maximum flaw size in solid material m
λs Thermal conductivity of solid material W/m.K
λg Thermal conductivity of the gas within the cells W/m.K
Cp,s Specific heat capacity of solid material J/kg.K
αs Thermal expansion coefficient of solid material mstrain/K
εr,s Dielectric constant of solid material Dimensionless
ρelec,s Electrical resistivity of solid material mW.cm
Dε,s Dielectric loss tangent of solid material Dimensionless

Top

Composites (Simple Bounds) model

Symbol Description Units (Metric)
f Volume fraction of reinforcement Dimensionless
(1-f) Volume fraction of matrix Dimensionless
ρr , ρm Density of reinforcement and matrix kg/m3
Young's modulus, yield strength, compressive strength of composite material Young's modulus, yield strength, compressive strength of composite material GN/m2
Er , Gr, Kr Young's modulus, shear modulus, bulk modulus of reinforcement GN/m2
Em , Gm, Km Young's modulus, shear modulus, bulk modulus of matrix GN/m2
νr, νm Poisson's ratio of reinforcement and matrix Dimensionless
σc,r , σy,r, σts,r , σflex,r Compression strength, yield strength, tensile strength, flexural strength of reinforcement MN/m2
σc,m , σy,m, σts,m , σflex,m Compression strength, yield strength, tensile strength, flexural strength of matrix MN/m2
Cp,r , Cp,m Specific heat capacity of reinforcement and matrix J/kg.K
αr , αm Thermal expansion coefficient of reinforcement and matrix mstrain/K
λr , λm Thermal conductivity of reinforcement and matrix W/m.K
λ1 , λ2 Thermal conductivity parallel and transverse to the fiber direction (Short Fiber Composite model only) W/m.K
ρelec,r , ρelec,m Electrical resistivity of reinforcement and matrix mW.cm
εr,r , εr,m Dielectric constant of reinforcement and matrix Dimensionless
De,r , Dε,m Dielectric loss tangent of reinforcement and matrix Dimensionless
(EE)r, (EE)m Embodied energy of reinforcement and matrix MJ/kg
(CF)r, (CF)m CO2 footprint of reinforcement and matrix kg/kg

Top

Controlled Thermal Expansion model

Symbol Description Units (Metric)
L Size of outer lattice unit cell m
D1 , D2 Diameter of outer and inner lattice struts m
A1 , A2 Cross sectional area of outer and inner lattice struts m2
θ Offset angle of the outer lattice °
l1 , l2 Size of lattice beam elements of outer and inner lattices m
ρ1 , ρ2 Density of the materials making up the outer and inner lattices kg/m3
α1 , α2 Coefficient of thermal expansion of the outer and inner lattice materials mstrain/K

Top

Cost

Part cost estimator

Symbol

Parameter definition

Source of data

Data type (used in calculation)

C

 final part cost Calculated Range

Cc

 capital cost of equipment Process datasheet Point (varies with part size and complexity)

Cm

 material price per unit mass Material datasheet Range

overhead rate User input Point

Cpp

 primary process cost and material cost Calculated Range

Csp

 secondary process cost Calculated Range

Crc

 credit for recycling waste Calculated Range

Ct

 tooling cost Process datasheet Point (varies with part size and complexity)

fpp

 primary process material utilization factor Process datasheet Point (maximum value of range)

fsp

 secondary process material utilization factor (= 1 - amount of scrap) User input Point

frm

 scrap material value as a percentage of material cost (Cm) User input Point

L

 load factor User input Point

l

 component length User input Point

m

 component mass User input Point

mi

initial material mass

Calculated

 Point

n

 batch size User input Point
[1] production rate (number / hour) Process datasheet Point (varies with part size and complexity)
nt[2] tool life Process datasheet Point (varies with part size and complexity)

two

 capital write-off time User input Point
ceiling smallest integer value above value in parentheses ({})

[1] If the processing method is a continuous (rather than batch) method, replace with

[2] If the processing method is a continuous (rather than batch) method, replace nt with nt/l

Top

Multi-layer Materials model

Symbol Description Units (Metric)
hi Thickness of layer i m
yi Distance between bottom of the multi-layer and top of layer i m
fi Relative volume fraction of layer i Dimensionless
I* Second moment of area of the structure per unit sample width m3
Ei Young's modulus of layer i GPa
ρi Density of layer i kg/m3
Kmax The maximum curvature that can be attained by the multi-layer m-1

n

 Total number of layers Dimensionless
σy,i Yield strength or compressive strength of layer i, dependent on layer's relative position to the neutral axis MPa
Ci Price per mass of layer i Currency/kg
Cp,i Specific heat capacity of layer i J/kg.K
αi Thermal expansion coefficient of layer i mstrain/K
λi Thermal conductivity of layer i W/m.K
ρelec,i Electrical resistivity of layer i mW.cm
εr,i Dielectric constant of layer i Dimensionless
De,i Dielectric loss tangent of layer i Dimensionless
(EE)i Embodied energy of layer i MJ/kg
(CF)i CO2 footprint of layer i kg/kg

Top

Sandwich Panels

Balanced Sandwich Panel model

Symbol Description Units (Metric)
t, c, d Face-sheet thickness, core thickness and overall panel thickness m
L, b Panel length and width m
ma Mass per unit area of the panel kg/m2
f = 2t/d Relative volume occupied by the face-sheet Dimensionless
(1-f) = c/d Relative volume occupied by the core Dimensionless
I Second moment of area m4
ρf, ρc Densities of face-sheet and core material kg/m3
equivalent density Equivalent density of panel kg/m3
Ef Young's modulus of the face-sheet GPa
Ec, Gc Young's modulus and shear modulus of the core GPa
σy,f Yield strength of face-sheet MPa
σy,c, σts,c, σc,c, σflex,c Yield strength, tensile strength, compressive strength and flexural strength of core MPa
equivalent in plane strength Equivalent in-plane strength of panel MPa
Cm,f , Cm,c Price per unit mass of face-sheet and core material Currency/kg
Cp,f, Cp,c Specific heat capacity of face-sheet and core material J/kg.K
αf, αc Thermal expansion coefficient of face-sheet and core material mstrain/K
λf , λc Thermal conductivity of face-sheet and core material W/m.K
ρelec,f , ρelec,c Electrical resistivity of face-sheet and core material mW.cm
εr,f , εr,c Dielectric constant of face-sheet and core material Dimensionless
De,f , Dε,c Dielectric loss tangent of face-sheet and core material Dimensionless
(EE)i Embodied energy of layer i MJ/kg
(CF)i CO2 footprint of layer i kg/kg

Top

Copyright © 2020, Granta Design, Cambridge, UK