# The NAFEMS Glossary

## ACCELERATION

The second time derivative of the displacement (the first time derivative of the velocity).
## ADAPTIVITY

See mesh adaptivity.
## ALGEBRAIC EIGENVALUE PROBLEM

The eigenvalue problem when written in the form of 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 (dS2 - dS02)/(2 dS2), where dS0 and dS are the undeformed and deformed lengths (see also Green’s strain).
## ALTERNATING PLASTICITY

Occurs in cyclic loading (q.v.) when there is a progressive increase in total strain with each cycle.
## 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.
## 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.
## 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.
## AREA COORDINATES

A special coordinate system that is used for defining shape functions for triangular and tetrahedral elements.
## ASPECT RATIOS

The ratio of the different element side or edge lengths, used for establishing amounts of distortion (q.v.).
## ASSEMBLY

The process of assembling the element matrices together to form the global matrix. Typically element stiffness matrices are assembled to form the complete stiffness matrix of the structure.
## ASSOCIATIVE PLASTICITY

A form of plasticity in which the yield function and the plastic potential are identical.
## 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.
## 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.
## AUTOMATIC MESH GENERATION

The process of generating a mesh of elements over the volume that is being analysed. There are two forms of automatic mesh generation: Free Meshing - Where the mesh has no structure to it. Free meshing generally uses triangular and tetrahedral elements. Mapped Meshing - Where large regions, if not all, of the volume is covered with regular meshes. This can use any form of element. Free meshing can be used to fill any shape. Mapped meshing can only be used on some shapes without elements being excessively distorted.
## AUTOMATIC NODE RENUMBERING BANDWITH PROFILE WAVEFRONT

The process of renumbering the nodes or elements to minimise the bandwidth, the profile or the wavefront of the assembled matrix. This renumbering is normally transparent to the user.
## AXISYMMETRIC ELEMENT

An element defined by rotating a cross-section about a centre line.
## AXISYMMETRIC THIN SHELL AXISYMMETRICAL THICK SHELL

An element forms an axisymmetric thin shell if a line element is rotated about an axis. An element forms an axisymmetric thick shell if a triangular or quadrilateral element is rotated about an axis.
## AXISYMMETRY

If a shape can be defined by rotating a cross-section about a line (e.g. a cone) then it is said to be axisymmetric. This can be used to simplify the analysis of the system. Such models are sometimes called two and a half dimensional since a 2D cross-section represents a 3D body.
## BANDWIDTH

The half bandwidth of a matrix is the maximum distance of any non-zero term in the matrix from the leading diagonal of the matrix. The bandwith for a symmetric matrix is then twice this.
## BARLOW POINTS

The set of Gauss integration points that give the best estimates of the stress for an element. For triangles and tetrahedra these are the full Gauss integration points. For quadrilateral and brick elements they are the reduced Gauss points.
## BASIS SPACE

When an element is being constructed it is derived from a simple regular shape in non-dimensional coordinates. The coordinates used to define the simple shape form the basis space. In its basis space a general quadrilateral is a 2x2 square and a general triangle is an isosoles triangle with unit side lengths.
## 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 ELEMENT

A line element that has both translational and rotational degrees of freedom. It represents both membrane and bending actions.
## BENDING

Bending behaviour is where the strains vary linearly from the centre line of a beam or centre surface of a plate or shell. There is zero strain on the centre line for pure bending. Plane sections are assumed to remain plane. If the stresses are constant normal to the centre line then this is called membrane behaviour.
## 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.
## BODY FORCE VECTOR

Mechanical loadings within the interior of the volume, typically inertia loadings in a stiffness analysis.
## 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.
## BOUNDARY ELEMENT BOUNDARY INTEGRAL

A method of solving differential equations by taking exact solutions to the field equations loaded by a point source and then finding the strengths of sources distributed around the boundary of the body required to satisfy the boundary conditions on the body.
## 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.
## BUBBLE FUNCTIONS

Element shape functions that are zero along the edges of the element. They are non-zero within the interior of the element.
## 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.
## BUCKLING (SNAP THROUGH)

The situation where the elastic stiffness of the structure is cancelled by the effects of compressive stress within the structure. If the effect of this causes the structure to suddenly displace a large amount in a direction normal to the load direction then it is classical bifurcation buckling. If there is a sudden large movement in the direction of the loading it is snap through buckling.
## 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
## 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.
## CENTRAL DIFFERENCE METHOD

A method for numerically integrating second order dynamic equations of motion. It is widely used as a technique for solving non-linear dynamic problems.
## CHARACTERISTIC VALUE

Same as the eigenvalue.
## CHARACTERISTIC VECTOR

Same as the eigenvector.
## CHOLESKY FACTORISATION (SKYLINE)

A method of solving a set of simultaneous equations that is especially well suited to the finite element method. It is sometimes called a skyline solution. Choose to optimise the profile of the matrix if a renumbering scheme is used.
## 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.
## COLUMN VECTOR (COLUMN MATRIX)

An nx1 matrix written as a vertical string of numbers. It is the transpose of a row vector.
## COMPATIBILITY EQUATIONS

Compatibility is satisfied if a field variable, typically the structural displacement, which is continuous before loading is continuous after loading. For linear problems the equations of compatibility must be satisfied. Nonlinearity in or non-satisfaction of, the compatibility equations leads to cracks and gaps in the structure. For finite element solutions compatibility of displacement is maintained within the element and across element boundaries for the most reliable forms of solution.
## COMPATIBILITY OF STRAINS

Compatibility of strain is satisfied if strains that are continuous before loading are continuous after. Admin
## COMPLETE DISPLACEMENT FIELD

When the functions interpolating the field variable (typically the displacements) form a complete n'th order polynomial in all directions.
## 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.
## 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.
## COMPOSITE MATERIAL

A material that is made up of discrete components, typically a carbon-epoxy composite material or a glass-fibre material. Layered material and foam materials are also forms of composite materials.
## 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.
## CONDENSATION STATIC CONDENSATION MODAL CONDENSATION

The reduction of the size of a problem by eliminating (condensing out) some degrees of freedom. For static condensation the elimination process is based upon static considerations alone. In more general condensation it can include other effects, typically model condensation includes both static and dynamic effects.
## CONDITION NUMBER

The ratio of the highest eigenvalue to the lowest eigenvalue of a matrix. The exponent of this number gives a measure of the number of digits required in the computation to maintain numerical accuracy. The higher the condition number the more chance of numerical error and the slower the rate of convergence for iterative solutions.
## CONDITIONAL STABILITY UNCONDITIONAL STABILITY

Any scheme for numerically integrating dynamic equations of motion in a step by step form is conditionally stable if there is a maximum timestep value that can be used. It is unconditionally stable (but not necessarily accurate) if any length of time step can be used.
## 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.
## CONGRUENT TRANSFORMATION

A transformation of the coordinate system of the problem that preserves the symmetry of the system matrices.
## CONJUGATE GRADIENT METHOD

A method for solving simultaneous equations iteratively. It is closely related to the Lanczos method for finding the first few eigenvalues and eigenvectors of a set of equations.
## 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.
## CONSISTENT DISPLACEMENTS AND FORCES

The displacements and forces act at the same point and in the same direction so that the sum of their products give a work quantity. If consistent displacements and forces are used the the resulting stiffness and mass matrices are symmetric.
## CONSISTENT TANGENT STIFFNESS METHOD

A technique in plasticity analysis using stiffnesses at each iteration that accurately incorporates the current state of plasticity.
## CONSTANT STRAIN CONSTANT STRESS

For structural analysis an element must be able to reproduce a state of constant stress and strain under a suitable loading to ensure that it will converge to the correct solution. This is tested for using the patch test.
## CONSTITUTIVE EQUATION

A description of any linear or non-linear material behaviour law, usually relating strain, stress and temperature.
## CONSTITUTIVE RELATIONSHIPS

The equations defining the material behaviour for an infinitesimal volume of material. For structures these are the stress-strain laws and include Hookes law for elasticity and the Prandle-Reuss equations for plasticity.
## 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.).
## CONSTRAINT EQUATIONS (MULTI POINT CONSTRAINTS)

If one group of variables can be defined in terms of another group then the relationship between the two are constraint equations. Typically the displacements on the face of an element can be constrained to remain plane but the plane itself can move.
## CONSTRAINTS

Fixed relationships between the basic degrees of freedom in a finite element model.
## CONTACT ELEMENTS/GAP ELEMENTS

Elements, as lines or areas, used to model states of contact between surfaces.
## CONTACT INSTABILITY

This occurs in contact analysis when instabilities are generated due to local mesh density and hourglassing. They can cause convergence problems.
## CONTACT PROBLEMS

A contact problem occurs when two bodies that are originally apart can come together, or two bodies that are originally connected can separate.
## CONTINUOUS MASS MODELS

The system mass is distributed between the degrees of freedom in a kinematically equivalent manner. The mass matrix is not diagonal.
## CONTINUOUS MODELS

The model is defined in terms of partial differential equations rather than in finite degree of freedom matrix form.
## 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.
## CONTOUR PLOTTING

A graphical representation of the variation of a field variable over a surface. A contour line is a line of constant value for the variable. A contour band is an area of a single colour for values of the variable within two limit values.
## 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.
## 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.
## 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.
## CONVERGENCE REQUIREMENTS

For a structural finite element to converge as the mesh is refined it must be able to represent a state of constant stress and strain free rigid body movements exactly. There are equivalent requirements for other problem types.
## CONVOLUTION INTEGRAL (DUHAMEL INTEGRAL)

The integral relating the dynamic displacement response of the structure at any time t to the forces applied before this time.
## COORDINATE SYSTEM

The set of displacements used to define the degrees of freedom of the system.
## CORRESPONDING FORCES AND DISPLACEMENTS

A force and a displacement are said to correspond if they act at the same point and in the same direction. Forces and translational displacements can correspond as can moments and rotations. Corresponding forces and displacements can be multiplied together to give a work quantity. Using corresponding forces and displacements will always lead to a symmetric stiffness matrix.
## 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.
## 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.
## CRACK ELEMENT (CRACK TIP ELEMENT)

An element that includes special functions to model the stress field at the tip of a crack. This is commonly achieved by using quadratic elements with mid side nodes at the quarter chord points.
## 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.
## 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 PROPAGATION (FRACTURE MECHANICS)

The process by which a crack can propagate through a structure. It is commonly assumed that a crack initiates when a critical value of stress or strain is reached and it propagates if it can release more than a critical amount of energy by the crack opening.
## 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.
## 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.
## 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.
## 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.
## CRANK-NICHOLSON SCHEME

A method for numerically integrating first order dynamic equations of motion. It is widely used as a technique for solving thermal transient problems.
## CREEP LAWS

The laws that govern time dependent creep, based on simple experimental tests. Typical laws are those of Norton, Prandtl, and Bailey.
## CREEP STRAIN

Irrecoverable permanent strain due to time dependent creep.
## CRITICAL DAMPING

The damping value for which the impulse response is just oscillatory.
## CRITICAL ENERGY RELEASE

This is a material property defining the minimum energy that a propagating crack must release in order for it to propagate. Three critical energies, or modes of crack propagation, have been identified. Mode 1 is the two surfaces of the crack moving apart. Mode 2 is where the two surfaces slide from front to back. Mode 3 is where the to surfaces slide sideways.
## 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.
## 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.
## CRITICALLY DAMPED SYSTEM CRITICAL DAMPING

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.
## CYCLIC LOADING

Loads that repeatedly oscillate between maximum and minimum values over time.
## CYCLIC SYMMETRY

A generalisation of axisymmetry. The structure is composed of a series of identical sectors that are arranged circumferentially to form a ring. A turbine disc with blades attached is a typical example.
## 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.
## DAMPED EIGENVALUES

Same as complex eigenvalues.
## DAMPED EIGENVECTORS

Same as complex eigenvectors.
## DAMPED NATURAL FREQUENCY

The frequency at which the damped system vibrates naturally when only an initial disturbance is applied.
## DAMPING

Any mechanism that dissipates energy; important in dynamics analysis.
## DAMPING FACTOR (DECAY FACTOR)

The damping factor is the ratio of the actual damping to the critical damping. It is often specified as a percentage. If the damping factor is less than one then the system can undergo free vibrations. The free vibrations will decay to zero with time. If the damping factor is greater than one then the decay is exponential and no vibrations occur. For most structures the damping factor is very small.
## DAMPING FACTOR/RATIO

The ratio of the viscous damping coefficient to the critical damping value.
## DEGENERATE ELEMENTS

Elements that are defined as one shape in the basis space but they are a simpler shape in the real space. A quadrilateral can degenerate into a triangle. A brick element can degenerate into a wedge, a pyramid or a tetrahedron. Degenerate elements should be avoided in practice.
## DEGREES OF FREEDOM

The number of equations of equilibrium for the system. In dynamics, the number of displacement quantities which must be considered in order to represent the effects of all of the significant inertia forces.
## DELAMINATION

The separation of layers of composites under stress.
## DET(J) DET J

The Jacobian matrix is used to relate derivatives in the basis space to the real space. The determinant of the Jacobian - det(j) - is a measure of the distortion of the element when mapping from the basis to the real space.
## DETERMINISTIC ANALYSIS

The applied loading is a known function of time.
## 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.
## DEVIATORIC STRESS

A measure of stress where the hydrostatic stress has been subtracted from the actual stress. Material failures that are flow failures (plasticity and creep) fail independently of the hydrostatic stress. The failure is a function of the deviatoric stress.
## DIAGONAL DECAY

When a matrix is factorised into a triangular form the ratio of a diagonal term in the factorised matrix to the corresponding term in the original matrix decreases in size as one moves down the diagonal. If the ratio goes to zero the matrix is singular and if it is negative the matrix is not positive definite. The diagonal decay can be used as an approximate estimate of the condition number of the matrix.
## DIAGONAL GENERALISED MATRIX

The eigenvectors of a system can be used to define a coordinate transformation such that, in these generalised coordinates the coefficient matrices (typically mass and stiffness) are diagonal
## DIE-AWAY LENGTH

If there is a stress concentration in a structure the high stress will reduce rapidly with distance from the peak value. The distance over which it drops to some small value is called the die-away length. A fine mesh is required over this die-away length for accurate stress results.
## DIRECT INTEGRATION

The name for various techniques for numerically integrating equations of motion. These are either implicit or explicit methods and include central difference, Crank-Nicholson, Runge-Kutta, Newmark beta and Wilson theta.
## DIRECTION COSINES

The cosines of the angles a vector makes with the global x,y,z axes.
## DISCRETE CRACK MODEL

In non-linear concrete analysis, a model that attempts to follow individual cracks.
## DISCRETE PARAMETER MODELS (DISCRETISED APPROACH)

The model is defined in terms of an ordinary differential equation and the system has a finite number of degrees of freedom.
## DISPLACEMENT CONTROL

When displacements are selected as the controlling parameter in a non-linear solution (as opposed to load or time control).
## DISPLACEMENT METHOD (DISPLACEMENT SOLUTION)

A form of discrete parameter model where the displacements of the system are the basic unknowns.
## DISPLACEMENT PLOTS

Plots showing the deformed shape of the structure. For linear small deflection problems the displacements are usually multiplied by a magnifying factor before plotting the deformed shape.
## 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.
## DISPLACEMENT VECTOR

The nodal displacements written as a column vector.
## DISSIMILAR SHAPE FUNCTIONS INCOMPATIBLE SHAPE FUNCTIONS

If two connecting elements have different shape functions along the connection line they are said to be incompatible. This should be avoided since convergence to the correct solution cannot be guarantied.
## DISTORTION (ALSO CALLED SHAPE SENSITIVITY)

An indication of how much an element’s shape differs from the theoretical shape for that element type.
## DISTORTION ELEMENT DISTORTION

Elements are defined as simple shapes in the basis space, quadrilaterals are square, triangles are isosoles triangles. If they are not this shape in the real space they are said to be distorted. Too much distortion can lead to errors in the solution
## 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 EQUIVALENT STRESSES

An equivalent stress measure for friction materials (typically sand). The effect of hydrostatic stress is included in the equivalent stress.
## 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.
## 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.
## DYNAMIC ANALYSIS

An analysis that includes the effect of the variables changing with time as well as space.
## DYNAMIC CONTACT

The analysis of contacting surfaces when inertia effects cannot be ignored.
## DYNAMIC FLEXIBILITY (ALSO CALLED RECEPTANCE)

The factor relating the steady state displacement response of a system to a sinusoidal force input.
## 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.
## DYNAMIC MODELLING

A modelling process where consideration as to time effects in addition to spatial effects are included. A dynamic model can be the same as a static model or it can differ significantly depending upon the nature of the problem.
## DYNAMIC STIFFNESS MATRIX

If the structure is vibrating steadily at a frequency w then the dynamic stiffness matrix is (K+iwC-w2M).
## DYNAMIC STRESSES

Stresses that vary with time and space.
## DYNAMIC SUBSTRUCTURING

Special forms of substructuring used within a dynamic analysis. Dynamic substructuring is always approximate and causes some loss of accuracy in the dynamic solution.
## 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.
## 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.
## EIGENVALUE PROBLEM

Problems that require calculation of eigenvalues and eigenvectors for their solution. Typically solving free vibration problems or finding buckling loads.
## 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 LATENT ROOTS CHARACTERISTIC VALUES

The roots of the characteristic equation of the system. If a 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. There are other problems that require the solution of the eigenvalue problem, the buckling loads of a structure are eigenvalues. Latent roots and characteristic values are synonyms for eigenvalues.
## EIGENVECTORS

The displaced shape that corresponds to the eigenvalues. If the structure is excited at a resonant frequency then the shape that it adopts is the mode shape corresponding to the eigenvalue.
## EIGENVECTORS LATENT VECTORS NORMAL MODES

The displacement shape that corresponds to the eigenvalues. If the structure is excited at a resonant frequency then the shape that it adopts is the mode shape corresponding to the eigenvalue. Latent vectors and normal modes are the same as eigenvectors.
## ELASTIC FOLLOW-UP

A structural phenomenon in which creep strain concentrates in rapidly creeping regions which are also relatively stiff; also analogously in plasticity.
## ELASTIC FOUNDATION

If a structure is sitting on a flexible foundation the supports are treated as a continuous elastic foundation. The elastic foundation can have a significant effect upon the structural response.
## ELASTIC STIFFNESS

If the relationship between loads and displacements is linear then the problem is elastic. For a multi-degree of freedom system the forces and displacements are related by the elastic stiffness matrix.
## 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 FIELDS

Electro-magnetic and electro-static problems form electric field problems.
## ELEMENT

In the finite element method the geometry is divided up into elements. Each element has nodes associated with it. The behaviour of the element is defined in terms of the freedoms at the nodes.
## ELEMENT ASSEMBLY

Individual element matrices have to be assembled into the complete stiffness matrix. This is basically a process of summing the element matrices. This summation has to be of the correct form. For the stiffness method the summation is based upon the fact that element displacements at common nodes must be the same.
## ELEMENT STRAINS ELEMENT STRESSES

Stresses and strains within elements are usually defined at the Gauss points (ideally at the Barlow points) and the node points. The most accurate estimates are at the reduced Gauss points (more specifically the Barlow points). Stresses and strains are usually calculated here and extrapolated to the node points.
## ELEMENT TYPES

A formal definition of individual element formulations.
## 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 METHODS HAMILTONS PRINCIPLE

Methods for defining equations of equilibrium and compatibility through consideration of possible variations of the energies of the system. The general form is Hamiltons principle and sub-sets of this are the principle of virtual work including the principle of virtual displacements (PVD) and the principle of virtual forces (PVF).
## ENERGY RELEASE RATE

See strain energy release rate.
## ENGINEERING NORMALISATION MATHEMATICAL NORMALISATION

Each eigenvector (mode shape or normal mode) can be multiplied by an arbitrary constant and still satisfy the eigenvalue equation. Various methods of scaling the eigenvector are used Engineering normalisation - The vector is scaled so that the largest absolute value of any term in the eigenvector is unity. This is useful for inspecting printed tables of eigenvectors. Mathematical normalisation - The vector is scaled so that the diagonal modal mass matrix is the unit matrix. The diagonal modal stiffness matrix is the system eigenvalues. This is useful for response calculations. ## ENGINEERING SIMULATION

The use of numerical, physical or logical models of systems and scientific problems in predicting their response to different physical conditions.
## ENGINEERING STRAIN (ALSO CALLED NOMINAL STRAIN)

The ratio of the change in length over a given length to the original length.
## 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.
## EQUILIBRIUM EQUATIONS

Internal forces and external forces must balance. At the infinitesimal level the stresses and the body forces must balance. The equations of equilibrium define these force balance conditions.
## EQUILIBRIUM FINITE ELEMENTS

Most of the current finite elements used for structural analysis are defined by assuming displacement variations over the element. An alternative approach assumes the stress variation over the element. This leads to equilibrium finite elements.
## EQUIVALENT MATERIAL PROPERTIES

Equivalent material properties are defined where real material properties are smeared over the volume of the element. Typically, for composite materials the discrete fibre and matrix material properties are smeared to give average equivalent material properties.
## EQUIVALENT STRAIN

See effective strain.
## EQUIVALENT STRESS

See effective stress.
## ERGODIC PROCESS

A random process where any one sample record has the same characteristics as any other record.
## EULERIAN FORMULATION

A geometrically non-linear formulation where the equilibrium conditions are evaluated in the deformed configuration (q.v. also Lagrangian formulation).
## EULERIAN METHOD LAGRANGIAN METHOD

For non-linear large deflection problems the equations can be defined in various ways. If the material is flowing though a fixed grid the equations are defined in Eulerian coordinates. Here the volume of the element is constant but the mass in the element can change. If the grid moves with the body then the equations are defined in Lagrangian coordinates. Here the mass in the element is fixed but the volume changes.
## EXACT SOLUTIONS

Solutions that satisfy the differential equations and the associated boundary conditions exactly. There are very few such solutions and they are for relatively simple geometries and loadings.
## EXPLICIT METHODS IMPLICIT METHODS

These are methods for integrating equations of motion. Explicit methods can deal with highly non-linear systems but need small steps. Implicit methods can deal with mildly non-linear problems but with large steps.
## 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.
## EXTRAPOLATION INTERPOLATION

The process of estimating a value of a variable from a tabulated set of values. For interpolation values inside the table are estimated. For extrapolation values outside the table are estimated. Interpolation is generally accurate and extrapolation is only accurate for values slightly outside the table. It becomes very inaccurate for other cases.
## FACETED GEOMETRY

If a curved line or surface is modelled by straight lines or flat surfaces then the modelling is said to produce a faceted geometry.
## 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.
## FAST FOURIER TRANSFORM

A method for calculating Fourier transforms that is computationally very efficient.
## FIELD PROBLEMS

Problems that can be defined by a set of partial differential equations are field problems. Any such problem can be solved approximately by the finite element method.
## FINITE DIFFERENCES

A numerical method for solving partial differential equations by expressing them in a difference form rather than an integral form. Finite difference methods are very similar to finite element methods and in some cases are identical.
## FINITE VOLUME METHODS

A technique related to the finite element method. The equations are integrated approximately using the weighted residual method, but a different form of weighting function is used from that in the finite element method. For the finite element method the Galerkin form of the weighted residual method is used.
## FLEXIBILITY MATRIX FORCE METHOD

The conventional form of the finite element treats the displacements as unknowns which leads to a stiffness matrix form. Alternative methods treating the stresses (internal forces) as unknowns leads to force methods with an associated flexibility matrix. The inverse of the stiffness matrix is the flexibility matrix.
## FLOW RULE

Used in plasticity to define a relationship between the plastic strain increment and the stress increment (q.v. also normality rule).
## FLUIDITY PARAMETER

In elastic-viscoplastic analysis, a parameter used in the evaluation of the viscoplastic strain rate.
## FOLLOWER FORCES

Forces that change their direction to follow geometric deformation during a large deformation analysis.
## FORCED RESPONSE

The dynamic motion resulting from a time varying forcing function.
## FORCING FUNCTION

The dynamic forces that are applied to a system.
## FORCING FUNCTIONS

The dynamic forces that are applied to the system.
## FOURIER EXPANSIONS FOURIER SERIES

Functions that repeat themselves in a regular manner can be expanded in terms of a Fourier series.
## 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.
## 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.
## 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.
## FRAMEWORK ANALYSIS

If a structure is idealised as a series interconnected line elements then this forms a framework analysis model. If the connections between the line elements are pins then it is a pin-jointed framework analysis. If the joints are rigid then the lines must be beam elements.
## FREE VIBRATION

The dynamic motion that results from specified initial conditions. The forcing function is zero.
## 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.
## FRONTAL SOLUTION WAVEFRONT SOLUTION

A form of solving the finite element equations using Gauss elimination that is very efficient for the finite element form of equations.
## 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.
## GAP ELEMENTS

See contact elements.
## GAUSS POINT EXTRAPOLATION GAUSS POINT STRESSES

Stresses calculated internally within the element at the Gauss integration points are called the Gauss point stresses. These stresses are usually more accurate at these points than the nodal points.
## 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.
## GAUSSIAN ELIMINATION

A form of solving a large set of simultaneous equations. Within most finite element systems a form of Gaussian elimination forms the basic solution process.
## GAUSSIAN INTEGRATION GAUSSIAN QUADRATURE

A form of numerically integrating functions that is especially efficient for integrating polynomials. The functions are evaluated at the Gauss points, multiplied by the Gauss weights and summed to give the integral.
## GENERALISED COORDINATES

A set of linearly independent displacement coordinates which are consistent with the constraints and are just sufficient to describe any arbitrary configuration of the system. Generalised coordinates are usually patterns of displacements, typically the system eigenvectors.
## GENERALISED MASS

The mass associated with a generalised displacement.
## GENERALISED STIFFNESS

The stiffness associated with a generalised displacement.
## GEOMETRIC STIFFNESS STRESS STIFFNESS

The component of the stiffness matrix that arises from the rotation of the internal stresses in a large deflection problem. This stiffness is positive for tensile stresses and negative for compressive stresses. If the compressive stresses are sufficiently high then the structure will buckle when the geometric stiffness cancels the elastic stiffness.
## GEOMETRICAL ERRORS

Errors in the geometrical representation of the model. These generally arise from the approximations inherent in the finite element approximation.
## GLOBAL STIFFNESS MATRIX

The assembled stiffness matrix of the complete structure.
## 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 (dS2 - dS02)/(2 dS02), where dS0 and dS are the undeformed and deformed lengths (see also Almansi strain).
## GRID

A term used in CFD for the connecting lines between nodes, equivalent to a mesh in finite element methods.
## GROSS DEFORMATIONS

Deformations sufficiently high to make it necessary to include their effect in the solution process. The problem requires a large deflection non-linear analysis.
## GROSS YIELDING

In elastic-plastic analysis, where widespread plasticity exists.
## GUARD VECTORS

The subspace iteration (simultaneous vector iteration) method uses extra guard vectors in addition to the number of vectors requested by the user. These guard the desired vectors from being contaminated by the higher mode vectors and speed up convergence.
## GUYAN REDUCTION METHOD

A method for reducing the number of degrees of freedom in a dynamic analysis. It is based upon a static approximation and always introduces some error in the computed dynamic solution. The error depends upon the choice of master freedoms.
## GYROSCOPIC FORCES

Forces arising from Coriolis acceleration. These can destabilise a dynamic response and cause whirling.
## 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.
## HARDENING STRUCTURE

A structure where the stiffness increases with load.
## HARMONIC LOADING

A dynamic loading that is periodic and can be represented by a Fourier series.
## HEAT CONDUCTION

The analysis of the steady state heat flow within solids and fluids. The equilibrium balance between internal and external heat flows.
## HEAT TRANSFER

The transfer of heat energy from one system to another. Heat transfer deals with the rate at which such energy is transferred.
## HERMITIAN SHAPE FUNCTIONS

Shape functions that provide both variable and variable first derivative continuity (displacement and slope continuity in structural terms) across element boundaries.
## HIDDEN LINE REMOVAL

Graphical plots of models where non-visible mesh lines are not plotted.
## 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.
## HIGH ASPECT RATIO LOW ASPECT RATIO

The ratio of the longest side length of a body to the shortest is termed its aspect ratio. Generally bodies with high aspect ratios (long and thin) are more ill-conditioned for numerical solution than bodies with an aspect ratio of one.
## HOLONOMIC CONSTRAINTS

Constraints that can be defined for any magnitude of displacement.
## HOOKES LAW

The material property equations relating stress to strain for linear elasticity. They involve the material properties of Youngs modulus and Poisson ratio.
## HOURGLASS EFFECTS

Spurious element deformations due to zero energy modes (q.v.).
## HOURGLASS MODE

Zero energy modes of low order quadrilateral and brick elements that arise from using reduced integration. These modes can propagate through the complete body.
## H-REFINEMENT P-REFINEMENT

Making the mesh finer over parts or all of the body is termed h-refinement. Making the element order higher is termed p-refinement.
## HYBRID COMPOSITE

A composite with two or more types of reinforcing fibres.
## HYBRID ELEMENTS

Elements that use stress interpolation within their volume and displacement interpolation around their boundary.
## 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).
## HYDROSTATIC STRESS

The stress arising from a uniform pressure load on a cube of material. It is the average value of the direct stress components at any point in the body.
## HYPERELASTICITY

A material which possesses an elastic potential function, known as the strain energy function, which is a scalar function of strain and whose derivatives with respect to each strain component gives the corresponding stress component.
## HYSTERETIC DAMPING

A model for the dissipation of energy in which the damping force is proportional to the amplitude of the displacement and opposes the velocity of motion.
## ILL-CONDITIONING ERRORS

Numerical (rounding) errors that arise when using ill-conditioned equations.
## ILL-CONDITIONING ILL-CONDITIONED EQUATIONS

Equations that are sensitive to rounding errors in a numerical operation. The numerical operation must also be defined. Equations can be ill-conditioned for solving simultaneous equations but not for finding eigenvalues.
## IMPLICIT SOLUTION SCHEME

An algorithm, usable in many types of non-linearity, whereby the solution for the next increment of time or load is obtained from the solution at the previous step and conditions from the current step (q.v. also explicit solution). It is used in both static and dynamic analyses.
## IMPULSE RESPONSE FUNCTION

The response of a system to an applied impulse.
## IMPULSE RESPONSE MATRIX

The matrix of all system responses to all possible impulses. It is always symmetric for linear systems. It is the inverse Fourier transform of the dynamic flexibility matrix.
## INCOMPRESSIBILITY

Straining with zero volumetric strain (i.e. no change in volume).
## INCOMPRESSIBLE FLOW

Flow where the density is not a function of pressure and so remains constant, with a Mach number (q.v.) below approximately 0.3.
## INCREMENTAL FORMULATION

The splitting up of applied load or time into small quantities (increments or steps) such that within each a meaningful converged solution can be conducted.
## INCREMENTAL SOLUTION

A solutions process that involves applying the loading in small increments and finding the equilibrium conditions at the end of each step. Such solutions are generally used for solving non-linear problems.
## INELASTIC MATERIAL BEHAVIOUR

A material behaviour where residual stresses or strains can remain in the body after a loading cycle, typically plasticity and creep.
## INERTANCE (ALSO CALLED ACCELERANCE)

The ratio of the steady state acceleration response to the value of the forcing function.
## INERTIA FORCE

The force that is equal to the mass times the acceleration.
## INITIAL BUCKLING

The load at which a structure first buckles.
## INITIAL STIFFNESS METHOD

A modified Newton-Raphson solution in which the initial linearly elastic, small displacement, stiffness matrix is used throughout the analysis.
## INITIAL STRAINS

The components of the strains that are non-elastic. Typically thermal strain and plastic strain.
## INTEGRATION BY PARTS

A method of integrating a function where high order derivative terms are partially integrated to reduce their order.
## INTERPOLATION FUNCTIONS SHAPE FUNCTIONS

The polynomial functions used to define the form of interpolation within an element. When these are expressed as interpolations associated with each node they become the element shape functions.
## ISOPARAMETRIC ELEMENTS

Elements in which the displacements and geometry variation within an element are represented by the same shape functions (q.v.).
## ISOTROPIC HARDENING

This occurs when, as plastic strains increase after initial yielding, the yield surface in principal stress coordinates expands uniformly about the origin while still maintaining its shape and orientation.
## ISOTROPIC MATERIAL

Materials where the material properties are independent of the co-ordinate system.
## JACOBI METHOD

A method for finding eigenvalues and eigenvectors of a symmetric matrix.
## JACOBIAN MATRIX

A square matrix relating derivatives of a variable in one coordinate system to the derivatives of the same variable in a second coordinate system. It arises when the chain rule for differentiation is written in matrix form.
## JACOBIANS

A mathematical quantity reflecting the distortion of an element from the theoretically perfect shape for that element type. It can be used as an element shape parameter (q.v.).
## J-INTEGRAL METHOD

Used in fracture mechanics to evaluate fracture parameters at a single crack tip calculated using an expression integrated along a path surrounding the tip.
## JOINTS

The interconnections between components. Joints can be difficult to model in finite element terms but they can significantly effect dynamic behaviour.
## KINEMATIC BOUNDARY CONDITIONS

The necessary displacement boundary conditions for a structural analysis. These are the essential boundary conditions in a finite element analysis.
## KINEMATIC HARDENING

This occurs when, as plastic strains increase after initial yielding, the yield surface in principal stress coordinates translates as a rigid body while maintaining its initial shape and orientation.
## KINEMATICALLY EQUIVALENT FORCES (LOADS)

A method for finding equivalent nodal loads when the actual load is distributed over a surface of a volume. The element shape functions are used so that the virtual work done by the equivalent loads is equal to the virtual work done by the real loads over the same virtual displacements. This gives the most accurate load representation for the finite element model. These are the non-essential stress boundary conditions in a finite element analysis.
## KINEMATICALLY EQUIVALENT LOADS

The point loads that are applied at the nodes of an element to represent a distributed load and that have been derived analytically to give the same work done as the distributed load.
## KINEMATICALLY EQUIVALENT MASS

If the mass and stiffness are defined by the same displacement assumptions, then a kinetically equivalent mass matrix is produced. This is not a diagonal (lumped) mass matrix.
## KINETIC ENERGY

The energy stored in a system arising from its velocity. In some cases, it can also be a function of the structural displacements.
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