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# Glossary and Guide to Technical Terms

This Glossary covers those terms that are thought to be of importance in the context of the present Guide. Most of these entries appear in glossaries from the various NAFEMS publications. Selected terms from benchmarks and other types of publication are also incorporated.

For each entry, the publication(s) in which they are described are given in italics, subdivided as Texts and Benchmarks as appropriate. Owing to the general nature and large volume of content in the Conference Proceedings, they are not included in these listings.

Two NAFEMS publications are dedicated Glossaries in their own rights:

HT01 , **Baguley, D, and Hose, D R,** How To Understand Finite Element Jargon, 1994.

HT26 , **De Souza, A,** How To Understand Computational Fluid Dynamics Jargon, 2003.

These publications provide a very extensive coverage of finite element related terms for solids and fluids, respectively. In the list below, the text HT01 has not been included in the Texts list since a HT01 is itself a glossary. A similar comment applies to HT26 , although that reference has been included in the Texts list for terms relating specifically to Computational Fluid Dynamics (CFD).

## The Glossary of Technical Terms

**Adaptivity** : see **mesh adaptivity** .

**Algebraic eigenvalue problem** : the eigenvalue problem can be written in the form (K-λM)u=0, which is: stiffness times mode shape minus eigenvalue times mass times mode shape is equal to zero. It is the form that arises naturally from a discrete parameter model in free vibration.

**Almansi strain** : strain defined in the deformed state as changes in squared length per twice the new squared length. It is given by (dS^{2} - dS_{0} ^{2} )/(2 dS^{2} ), where dS_{0} and dS are the undeformed and deformed lengths (see also Green’s strain).

Texts R0018 , R0072 , R0080 ,

Benchmarks P10 , R0065 .

**Alternating plasticity** : occurs in cyclic loading (q.v.) when there is a progressive increase in total strain with each cycle.

Text R0080 ,

Benchmarks R0026 , R0030 , R0049 .

**Anisotropy** : a material where the response to load depends on the direction within the material. In general, 21 independent constants are required to relate stress and strain.

Texts HT07 , HT25 , HT27 , R0091 ,

Benchmark R0003 .

**Arbitrary Lagrangian Eulerian mesh updating** : an automatic mesh re-zoning algorithm where a manual decision is replaced by regular re-zoning operations carried out at fixed time increments or number of calculation cycles.

Text R0085 .

**Arc length method** : a non-linear iterative technique used to solve non-linear problems at or near limit points, where there is a change in sign of the slope of the load-displacement curve.

Texts HT19 , R0018 , R0072 , R0080 , R0095 ,

Benchmark P03 .

**Aspect ratios** : the ratio of the different element side or edge lengths, used for establishing amounts of distortion (q.v.).

Texts FEP , HT07 , HT28 , P12 ,

Benchmark P08 .

**Associative plasticity** : a form of plasticity in which the yield function and the plastic potential are identical.

Texts R0018 , R0072 , R0080 , R0085 .

**Augmented Lagrangian method** : a combination of the penalty function and Lagrange multiplier methods (q.v.). Used in contact analysis, where the contact force is defined in terms of the Lagrange multiplier plus a penalty stiffness term.

Text R0025 ,

Benchmark R0081 .

**Automatic load/time incrementation** : a method for automatic incrementation in an applied load or time incremental-iterative solution process, i.e. the increment sizes are not specified by the analyst.

Text R0018 .

**Bauschinger effect** : observed in plasticity when, after initial tensile loading into the plastic region, the yield stress in compression is less than the equivalent value in tension.

**Beam properties** : geometrical and material quantities relevant to beams and which take advantage of the one dimensional nature of beams. The properties include moment of inertia, second moment of inertia, torsional rigidity and flexural rigidity.

**Beam theories** : theoretical formulations which take advantage of the one dimensional nature of beams. The two main theories are Euler-Bernoulli and Timoshenko.

**Bifurcation** : occurs on a non-linear load-displacement curve as the load path forks into two or more solution paths that satisfy equilibrium. Only one path is stable, the others being unstable.

Texts R0018 , R0072 , R0080 , R0085 ,

Benchmarks P03 , P10 , R0065 .

**Boundary conditions** : prescribed degrees of freedom and other quantities within a finite element model, which represent the physical model and are required to produce a unique solution for any type of applied loading.

Texts FEP , GDS, HT07 , HT28 , HT34 , R0098 .

**Brittle Fracture** : this is the type of fracture occurring for a crack in a material whose behaviour is described as brittle, when any plastic deformation is very limited so that fracturing occurs without significant prior deformation. This is typified by glassy materials and metals at temperatures below the range of the brittle-ductile transition temperature.

Texts HT17 , HT18 , R0018 ,

Benchmarks R0020 , R0028 .

**Buckling** : buckling is a geometric instability, generally caused by compressive forces in thin-sectioned bodies. It can be analysed as a special case of geometric non-linearity using eigenvalue analysis.

Texts FEP, HT19 , R0018 , R0072 , R0080 , R0095 ,

Benchmarks P03 , R0065 .

**Cam-Clay model** : a model describing the behaviour of clay-type soils, using a hardening/softening elastic-plastic constitutive law based on the critical state framework whose yield surface plots as a logarithmic curve.

Text HT27 .

**Cauchy stress** : see **true stress** .

**Cell** : a term used in CFD for a discrete area or volume over which the governing equations are integrated, equivalent to an element in finite element methods. The complete group of cells should define the domain under consideration.

**Closed-form displacement method** : for fracture mechanics, a special form of displacement substitution that only uses the calculated values in the crack tip elements.

**Coefficient of viscous damping** : the constants of proportionality relating the velocities to the forces.

Text R0017 .

**Complex eigenvectors** : the eigenvectors of any damped system. For proportionally damped systems, they are the same as the undamped eigenvectors. For non-proportionally damped systems with damping in all modes less than critical they are complex numbers and occur as complex conjugate pairs.

Text R0017 .

**Complex eigenvalues** : the eigenvalues of any damped system. If the damping is less than critical they will occur as complex conjugate pairs even for proportionally damped systems. The real part of the complex eigenvalue is a measure of the damping in the mode and should always be negative. The imaginary part is a measure of the resonant frequency.

Text R0017 .

**Compressible flow** : flow in gaseous fluids where speeds are sufficiently high, causing significant fluid density changes. It typically occurs when the Mach number (q.v.) exceeds approximately 0.3.

**Conduction** : a mode of heat transfer in which the heat energy is transferred on a molecular scale with no movement of macroscopic particles (matter) relative to one another: described by Fourier’s law.

Text HT14 .

**Conservation of energy** : the energy entering or leaving a volume of fluid due to flow convection and conduction is balanced by the energy of the fluid volume over time and the dissipation due to viscous forces.

**Conservation of mass** : the condition that mass cannot be created or destroyed within a fluid flow system.

**Conservation of momentum** : the condition that the forces on a fluid in a certain volume equal the mass of that fluid multiplied by its acceleration, effectively Newton’s second law of motion.

**Conservative Load** : a load that always acts in a fixed direction regardless of the deformation of the body, for example, gravity.

Texts R0018 , R0072 , R0080 ,

Benchmark R0065 .

**Consistent tangent stiffness method** : a technique in plasticity analysis using stiffnesses at each iteration that accurately incorporates the current state of plasticity.

**Constitutive equation** : a description of any linear or non-linear material behaviour law, usually relating strain, stress and temperature.

Texts FEP , HT19 , R0018 , R0072 , R0080 .

**Constrained methods** : non-linear solution procedures in which the solution is constrained to follow a certain path during the iteration process, e.g. arc length methods (q.v.).

Text R0018 .

**Constraints** : fixed relationships between the basic degrees of freedom in a finite element model.

Texts FEP , GDS, HT07 , HT28 .

**Contact elements/gap elements** : elements, as lines or areas, used to model states of contact between surfaces.

Texts FEP , HT15 , R0018, R0025 , R0072 .

**Contact instability** : this occurs in contact analysis when instabilities are generated due to local mesh density and hourglassing. They can cause convergence problems.

Text R0085 .

**Continuous mass models** : the system mass is distributed between the degrees of freedom in a kinematically equivalent manner. The mass matrix is not diagonal.

Text R0017 .

**Continuum region element (CRE) method** : a single element test where the element is defined within a region where there is a known stress field. Point loads and nodal displacements can then be calculated and applied over the element, whose shape can vary at will, to test the element’s response.

Text HT28 ,

Benchmarks P01 , R0012 .

**Convected coordinate formulation (** also called **co-rotational formulation)** : a geometrically non-linear formulation in which a local cartesian coordinate system is attached to the element and is allowed to continuously translate and rotate with the element during deformation.

**Convection** : a mode of heat transfer between a fluid and solid boundary. The heat energy is transferred by the movement of macroscopic fluid particles.

Text HT14 .

**Convergence** : for any non-linear solution procedure, convergence is achieved when sufficient iterations within a given increment of time or load have produced an equilibrium state to within a given convergence criterion.

Texts HT19 , R0018 , R0072 , R0080 .

**Convergence criterion** : in a non-linear solution procedure, this specifies how to decide whether convergence has been achieved within a given increment of time or load.

Texts HT19 , R0018 , R0072 , R0080 .

**Coulomb damping (** also called **dry friction damping)** : a damping model in which the damping force is a constant and always opposes the velocity of motion.

**Coupled problems** : these occur when multiple geometric domains are to be linked or when different physical states are to be solved, in each case in a dependent manner.

Text R0018 .

**Crack closure work methods** : these calculate the energy release rate by two finite element calculations, calculating the point force needed to either open or close the crack over a short length after the first run, and equating this work done to the required energy change; several variants exist.

Text HT18 ,

Benchmarks R0020 , R0028 .

**Crack propagation** : the relatively steady growth of cracks, usually during the fatigue life of a product. It could also be due to non-linear material degradation such as ductile void growth and coalescence.

**Crack profile or front** : The sharp end of a crack inside a three dimensional body, which is a curve of known position and of finite length, and which can vary with time. Any two dimensional section cutting this crack profile will contain a part of the crack ending in a crack tip (q.v.), and is frequently studied to give simplified values of the fracture parameters of interest.

Texts HT17 , HT18 , R0018 ,

Benchmarks R0020 , R0028 , R0038 .

**Crack tip** : The sharp end of a crack inside a given two dimensional body, at a point whose position is known and which may move over time.

Texts HT17 , HT18 , R0018 ,

Benchmarks R0020 , R0028 , R0038 .

**Crack tip elements** : finite elements sited around crack tips, modified to contain displacement variations representing the singular strain fields that exist there, thereby giving greater accuracy than the standard polynomial variations.

Texts HT18 , R0018 ,

Benchmarks R0020 , R0028 , R0038 .

**Crack tip equations** : These are mathematical equations which are valid for elastic crack tip conditions, relating components of stress and displacement with local geometric position relative to the crack tip. The equations give the stress intensity factors.

Text HT18 .

**Crack tip opening displacement (CTOD)** : This is a measure of how much the crack tip opens up under load when significant plastic deformation occurs in that region. It is useful as a fracture parameter.

Texts HT18 , R0095 ,

Benchmark R0038 .

**Creep laws** : the laws that govern time dependent creep, based on simple experimental tests. Typical laws are those of Norton, Prandtl, and Bailey.

Texts HT19 , HT21 , R0018 , R0072 , R0080 , R0091 ,

Benchmarks R0027 , R0049 .

**Creep strain** : irrecoverable permanent strain due to time dependent creep.

Texts HT19 , HT21 , R0018 , R0072 , R0080 ,

Benchmarks R0027 , R0049 .

**Critical damping** : the damping value for which the impulse response is just oscillatory.

**Critically damped system** : the dividing line between under damped and over damped systems where the equation of motion has a damping value that is equal to the critical damping.

Text R0017 .

**Critical Values** : These are numerical quantities representing the various fracture parameters, at those levels of load that cause some relevant fracture event to happen. For example, the critical value of the stress intensity factor is the fracture toughness.

Texts HT17 , HT18 , R0018 ,

Benchmarks R0020 , R0028 , R0038 , R0084 .

**Cyclic loading** : loads that repeatedly oscillate between maximum and minimum values over time.

Texts R0018 , R0072 ,

Benchmark R0049 .

**Cyclic symmetry** : geometric repetition in the form of cyclic sectors that can be used to minimise mesh modelling.

**Damage tolerance** : a design and operational philosophy in which products are regularly inspected for damage, crack growth, etc., so that continued operation with the damage will not produce an imminent failure.

Text HT17 .

**Damping** : any mechanism that dissipates energy; important in dynamics analysis.

Texts FEP , HT36, R0017 , R0021 , R0023 , R0085 .

**Damping factor/ratio** : the ratio of the viscous damping coefficient to the critical damping value.

**Damped natural frequency** : the frequency at which the damped system vibrates naturally when only an initial disturbance is applied.

Text R0017 .

**Delamination** : the separation of layers of composites under stress.

**Deviatoric stress and strain** : represents the shear component of stress and strain, i.e. the remainder after deducting the hydrostatic component (q.v.). The deviatoric components govern plastic and creep flows, where there is change in shape but not of volume.

Texts FEP , HT19 , HT27 , R0018 , R0072 , R0080 , R0085 ,

Benchmarks P06 , R0049 .

**Dielectric material** : an electric insulator which exhibits some degree of polarisation when placed in an electric field.

Text HT39 .

**Discrete crack model** : in non-linear concrete analysis, a model that attempts to follow individual cracks.

**Displacement control** : when displacements are selected as the controlling parameter in a non-linear solution (as opposed to load or time control).

Texts HT19 , R0018 , R0072 , R0080 .

**Displacement substitution method** : a method of calculating the stress intensity factor at a given crack tip using the local displacements from FE analysis and known crack tip equations.

Text HT18 ,

Benchmarks R0020 , R0028 .

**Distortion (** also called **shape sensitivity)** : an indication of how much an element’s shape differs from the theoretical shape for that element type.

Texts FEP , HT07 , HT28 , P11, P14 ,

Benchmarks P01 , P08 , R0012 .

**Domain integrals** : used in fracture mechanics to evaluate fracture parameters at a crack tip, calculated using an expression integrated over an area inside a given path surrounding the tip (also called thick contours; the area form of the J-integral (q.v.)).

**Drucker-Prager yield criterion** : an elasto-plastic material model using both hydrostatic and deviatoric stresses, that is an inverted cone in principal stress space. Used in soil mechanics.

Texts HT19 , HT27 , R0018 , R0080 , R0085 , R0091 .

**Ductile Fracture** : This is the type of fracture occurring for a crack in a material whose behaviour is ductile, i.e. when plastic deformation is considerable. Such fracturing occurs, after some general plastic deformation as the load builds up, in metals at temperatures above the range of the brittle-ductile transition temperature.

Texts HT18 , R0018 ,

Benchmark R0038 .

**Dynamic contact** : the analysis of contacting surfaces when inertia effects cannot be ignored.

Text HT15 .

**Dynamic flexibility (** also called **receptance)** : the factor relating the steady state displacement response of a system to a sinusoidal force input.

Text R0017 .

**Dynamic flexibility matrix** : the matrix relating the complete set of steady state displacement responses to all possible sinusoidal force inputs. It is always symmetric for linear systems. It is the Fourier transform of the impulse response matrix.

Text R0017 .

**Dynamic stiffness matrix** : if the structure is vibrating steadily at a frequency w then the dynamic stiffness matrix is (K+iwC-w^{2} M).

**Effective strain (** also called **equivalent strain)** : A scalar quantity defined (usually as the von Mises strain) to represent the individual strain components at any reference point: also used for strain rates.

Texts HT19 , R0018 , R0080 ,

Benchmarks R0027 , R0049 .

**Effective stress (** also called **equivalent stress)** : A scalar quantity defined (usually as the von Mises stress, q.v.) to represent the individual stress components at any reference point.

Texts FEP , HT19 , HT27 , R0018 , R0072 , R0080 ,

Benchmarks R0027 , R0049 .

**Eigenvalues** : the roots of the characteristic equation of a dynamic system. If the system has n equations of motion then it has n eigenvalues. The square root of the eigenvalues are the resonant frequencies. These are the frequencies that the structure will vibrate at if given some initial disturbance with no other forcing. Eigenvalues are also used in buckling analysis, where they become buckling loads.

**Eigenvectors** : the displaced shape that corresponds to the eigenvalues (q.v.). For dynamics, if the structure is excited at a resonant frequency then the shape that it adopts is the mode shape corresponding to the eigenvalue. For buckling, the corresponding eigenvector is a buckling shape.

**Elastic follow-up** : a structural phenomenon in which creep strain concentrates in rapidly creeping regions which are also relatively stiff; also analogously in plasticity.

Text R0018 .

**Elastic unloading** : this can occur in regions of structures that have become plastic and then have their stresses reduced to become elastic again, with plastic strains remaining.

**Elastic-Plastic Fracture Mechanics** : see **post yield fracture mechanics.**

**Electric field** : a field existing around a body with an electric charge, when placed close to another charged body.

Text HT39 .

**Electric field intensity** : the gradient of the electric potential at any position in a direction normal to the local line of equipotential. It is also known as electric field strength since it measures the magnitude of a force acting on a charged body in an electric field.

Text HT39 .

**Electric flux** : a measure of the number of electric field lines between two bodies of different electric potential that pass through a virtual area.

Text HT39 .

**Electric flux density** : the amount of electric flux in a tube of unit cross-sectional area perpendicular to the direction of the flux.

Text HT39 .

**Electric permittivity** : a material property which measures the ability of the material to transmit electric field. This ability is related to how well the material is polarised in an electric field.

Text HT39 .

**Electric potential** : a measure of the strength of an electric charge at any point.

Text HT39 .

**Electromagnetism** : the physical phenomena and effects of combined electricity and magnetism.

Text HT39 .

**Electromotive force** : the force required to maintain a flow of electrons along a conducting medium, in the form of electric current.

Text HT39 .

**Electrostatic force** : the attractive or repulsive force between 2 charged bodies .

Text HT39 .

**Element types** : a formal definition of individual element formulations.

Texts FEP , GDS, HT07 , HT28 , R0085 , R0098 .

**Energy Difference Technique** : used in fracture mechanics to evaluate the potential energy release rate at a single crack tip from the potential energies of two finite element runs differing only by a small change in crack length.

**Energy release rate** : see **strain energy release rate** .

**Engineering strain (** also called **nominal strain)** : the ratio of the change in length over a given length to the original length.

Texts HT19 , R0018 , R0072 , R0080 ,

Benchmarks P03 , P10 , R0065 .

**Equilibrium** : the state of a loaded body when the internal stresses are in equilibrium with the externally applied loads, and which always has to be achieved in finite element algorithms.

Texts FEP , HT19 , R0018 , R0072 , R0080 .

**Equivalent strain** : see **effective strain** .

**Equivalent stress** : see **effective stress** .

**Eulerian formulation** : a geometrically non-linear formulation where the equilibrium conditions are evaluated in the deformed configuration (q.v. also Lagrangian formulation).

Texts FEP , HT19 , R0018 , R0072 , R0080 , R0085 ,

Benchmark P10 .

**Explicit solution scheme** : an algorithm, used in many time or load dependent analyses, whereby the solution for the next increment of time or load is obtained entirely from the solution and conditions at the previous step (q.v. also implicit solution). It is used in both static and dynamic analyses.

Texts FEP , HT19 , R0018 , R0072 , R0080 , R0085 , R0095 .

**Fail-safe** : a design philosophy in which products are designed in such a way that failures prior to the required operational life are not catastrophic.

Text HT17 .

**Fast Fourier transform** : a method for calculating Fourier transforms that is computationally very efficient.

Text R0017 .

**Flow rule** : used in plasticity to define a relationship between the plastic strain increment and the stress increment (q.v. also normality rule).

Texts HT19 , R0018 , R0072 , R0080 .

**Fluidity parameter** : in elastic-viscoplastic analysis, a parameter used in the evaluation of the viscoplastic strain rate.

Text R0018 .

**Follower forces** : forces that change their direction to follow geometric deformation during a large deformation analysis.

Texts R0018 , R0072 , R0080 , R0085 , R0095 .

**Forced response** : the dynamic motion resulting from a time varying forcing function.

**Forcing function** : the dynamic forces that are applied to a system.

**Fourier transform** : a method for finding the frequency content of a time varying signal. If the signal is periodic it gives the same result as the Fourier series.

**Fourier transform pair** : the Fourier transform and its inverse which, together, allow the complete system to be transformed freely in either direction between the time domain and the frequency domain.

Text R0017 .

**Fracture Parameters/Criteria** : These are numerical quantities which represent the conditions at a crack tip in a given geometry at a given load level, e.g. , CTOD.

Texts HT17 , HT18 , R0018 ,

Benchmarks R0020 , R0028 , R0038 .

**Fracture Toughness (** **)** : For a given material, thickness and temperature, fracture toughness is the critical value of the stress intensity factor needed for a crack to grow under monotonic loading.

Texts HT17 , HT18, R0018 ,

Benchmark R0084 .

**Free vibration** : the dynamic motion that results from specified initial conditions. The forcing function is zero.

Texts FEP , R0017 ,

Benchmarks P09 , P18 , R0015 .

**Frequency domain** : a structure’s forcing function and the consequent response is defined in terms of their frequency content. The inverse Fourier transform of the frequency domain gives the corresponding quantity in the time domain.

**Frictional/frictionless contact** : in contact analysis, the state of different surfaces coming into contact. Frictional is when the surfaces are sufficiently rough that friction is important and either sticking or slipping can occur. Frictionless is when the surfaces are assumed to be perfectly lubricated and so no friction occurs.

Texts HT15 , R0025 , R0072 ,

Benchmark R0081 .

**Froude number** : a fluid flow measure of the ratio of inertia forces to gravitational forces, typically used in free surface flows.

**Gap chattering** : this occurs in contact analysis when certain gaps repeatedly open and close. This is an effect of the contact algorithm and can cause convergence problems.

Benchmark R0081 .

**Gap elements** : see **contact elements** .

**Gauss points** : strategic locations within elements where numerical integration and stress evaluations are made. These vary over different element types and can differ depending on usage for both numerical integration and stress evaluation.

Texts FEP , HT07 , HT08 , HT17 , HT18 , HT19 , HT28 , R0018 , R0080 .

**Green’s strain** : a strain measure used in geometric non-linear analysis and defined, with reference to the original configuration, as the change in the squared length divided by twice the original squared length. It is given by (dS^{2} - dS_{0} ^{2} )/(2 dS_{0} ^{2} ), where dS_{0} and dS are the undeformed and deformed lengths (see also Almansi strain).

Texts HT19 , R0018 , R0072 , R0080 ,

Benchmarks P03 , P10 , R0065 .

**Grid** : a term used in CFD for the connecting lines between nodes, equivalent to a mesh in finite element methods.

**Gross yielding** : in elastic-plastic analysis, where widespread plasticity exists.

Texts HT19 , R0018 , R0071 , R0080 .

**Hardening** : in non-linear material behaviour, the change in the current yield stress as plastic or creep straining occurs, such as work and strain hardening in plasticity, and time and strain hardening in creep.

Texts FEP , HT19 , HT27 , R0018 , R0072 , R0080 , R0085 ,

Benchmarks P06 , R0049 .

**Heat transfer** : the transfer of heat energy from one system to another. Heat transfer deals with the rate at which such energy is transferred.

Texts HT14 , HT35.

**Hierarchical elements** : element families with varying shape function orders such that the stiffness matrix of each order contains the stiffness matrices of each of the lower ordered elements as sub-matrices.

Text P13 .

**Hourglass effects** : spurious element deformations due to zero energy modes (q.v.).

**Hybrid composite** : a composite with two or more types of reinforcing fibres.

Text HT25 .

**Hydrostatic (** also called **Volumetric) stress and strain** : is the average of the direct stress or strain components at any point of reference, ignoring the shear components. It causes change in volume but not change in shape of an element of material (q.v. also deviatoric stress and strain).