** Arc length method: **(also called the Riks method ) is an iterative technique used to overcome numerical problems at or near limit points in the force-displacement curve. it advances the solutions by a suitable arc length ( rather than a load or displacement increment ) measured along the load path itself.

** Aspect ratio: **of an element is the ratio of the largest to the smallest side of the element, used as a measure of the slenderness of the element. Large aspect ratios should in many cases be avoid if possible.

**Axisymmetric**** problems: **are formed by rotating a two-dimensional flat plane through 360° about a fixed axis (usually the z-axis). For an axisymmetric assumption to be valid, the geometry and all the variables must be axisymmetric.

** Banded matrix: **is a matrix with its non-zero coefficients placed in a ” band ” around the diagonal. Special solvers can be used for banded matrices.

** Bar elements: **are usually one-dimensional elements which can only transmit axial forces without any rotations. The degrees of freedom are translations at cache node.

** Bauschinger effect:** is observed when the yield stress in compression is less than the equivalent value in tension.

** Beam elements: **are elements that satisfy the beam bending equations, with degrees of freedom representing both the translational displacement and rotations. The applied loads on beam elements are shear forces and bending moments .

** Benchmarks: **are problems with known and reliable reference solutions, that can be used to test the accuracy of FE programs.

** Bifurcation: **may occur in geometrically non-linear problems , such as buckling problems, where one or more solutions paths that satisfy equilibrium intersect on the load-displacement curve.

** Brick elements: **are three-dimensional continuum elements in the shape of hexahedral.

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

** Buckling: **is a geometric instability usually caused by compressive forces. Buckling can be analyses as a special case of geometric non-linearity involving equations of the type classified mathematically as “eigenvalue” problems .

** Cartesian coordinates:** are defined with respect to orthogonal sets of axes, usually written (x,y,z). They are also called global axes.

** Cauchy stress: **(also called true stress ) is defined as the force over the current (instantaneous) area.

** Conduction:** is 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.

** Conservative load: **is a load which always applies in a fixed directions regardless of the deformation of the body. A typical example is a gravitational load , which always applies vertically.

** Constitutive equations: **are the relationships between stress and strain of any linear or non-linear material behavioral law, such as Hooke’s law which uses Young’s modulus and Poisson’s ratio for elastic behavior.

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

**Contact problems:**** **are non-linear problems in which two or more surfaces come into or out of contact. Points in the contact areas may stick or slip according to the prescribed coefficient of friction.

** Continuity: **is used to describe the continuous nature of displacements across elements boundaries, i.e. no step changes in displacement between adjacent elements. Stresses, however, are allowed to be discontinuous across element boundaries.

** Contour plots: **are color plots showing lines of constant values (usually stress or strain) over all or part of the body.

** Convergence: **of a solution occurs when either the mesh size and/or time step have been made sufficiently small, such that no further significant changes in the calculated results occur if they were made smaller still. In non-linear problems, convergence is usually used to indicate the accuracy and reliability of the iterative procedures.

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

** Creep: **is a time-dependent non-linear material behavior which usually occurs in metals at high temperatures

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

** Deformed shape: **is the shape a body takes after loading. A deformed shape FE mesh plot usually involves magnifying the actual displacements by multiplying the modal a suitable factor and adding them to the nodal coordinates.

** Degrees of freedom: **are the nodal variables, usually displacement components in the directions of the main axes, or temperatures for thermal problems. Non-displacement values, such as rotations, are also used as degrees of freedom, typically in beam, plate and shell elements.

**Deviatoric**** stress: **represents the shear component of the stress, i.e. the remainder of the stress after deducting the hydro static stress component. The deviatoric stress is responsible for the plastic flow and creep behavior through the shear component.

** Displacement control: **refers to problems where the displacement, rared than the load, at a specified node or a set of nodes is increased by a small increment. It is recommended for snap-through problems, because the load -displacement tangent becomes horizontal and the load must subsequently remain constant or decrease in order to follow the load-displacement curve.

** Energy formulation: **is used to derive the element stiffness matrix, utilizing the principle of minimum total potential energy or other energy considerations.

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

** Engineering stress: **(also called nominal stress) is defined as the force divided by the original unreformed area.

** Equilibrium equations:** are differential equations containing stresses. They are obtained by considering the equilibrium of forces over a small differential volume and relate them to the rate of change of stress over that volume.

** Equivalent stress/strain: **(also called effective stress/strain) is a scalar quantify defined to represent the individual stress or strain components at any point. It usually refers to the von Mises stress/strain.

** Explicit Euler method: **(also known as forward-difference) is a time-marching scheme in which the new value of the variable is arrived at by using the tangent at the previous time step. This method needs very small time steps to remain stable.

** Flow rule: **is used to define the constitutive relationship between the plastic strain increment and the stress increment.

** Follower load: **is a load which changes its directions during the deformation of the structure, such as an internal pressure in a vessel which changes its positions and directions to remain perpendicular to the surface as the vessel deforms.

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

** Free vibration:** is the dynamic motions that occurs where an external force is applied to the system to displace it from its equilibrium position, but is then removed.

**Front or frontal solution:**** **is a special technique used to solve the FE algebraic equations by introducing elements, one at a time, to the solver. It is used to reduce the amount of computer memory required.

** Functional:** (in FE analysis) is a function to be minimized in the variation or energy formulations. In stress analysis problems, the functional is the total potential energy of the structure.

**Gallerkin**** method: **is a special form of the weighted residual technique widely used in FE formulations. Special weighting function are used which are equal to the approximate (trial) solutions.

** Gap elements: **are used in contact problems to approximately impose the contact conditions by joining the nodes across the contact surfaces. The stiffness of the gap elements determines whether the nodes are in or out of contact.

** Gauss points: **are positions within an element where numerical integration and stress evaluations are made. For many elements types they are the locations where the stresses are most accurately evaluated.

** Gaussian elimination: **is a standard reliable technique of solving a system of linear equations.

** Gaussian quadrature/integration:** is a well established numerical algorithm used to perform numerical integration. This technique is widely used in FE programs to calculate the integrals needed to obtain the element stiffness matrix efficiently.

** Geometric non-linear: **occurs when the changes in the geometry of a structure due to its displacement under load should be takes into account in analyzing its behavior. Geometric non-linearity problems can involve large or small deformations and strains.

** Global axes: **are the reference axes for the structure, usually defined by (x,y,z) Cartesian coordinates.

**Hermitia**** elements: **are described by Hermitian polynomials. They usually refer to beam, plate and shell elements, where the variables include rotations (slopes) as well as displacements.

**Hertzian contact**: Problems are elastic contact problems in which the contacting surfaces are generally curved (with no sharp edges or concerns) and frictionless. The contact area is assumed to be small compared to the principal radii of the undeformed surfaces.

**Hourglass mode**: Is a spurious state of element deformation due to zero energy modes. The distorted shape gives zero stiffness when the element is integrated. This arises when reduced integration is used in certain circumstances and the strains are zero at the integration points.

**Hybrid elements**: Are elements in which stresses as well as displacements are used as the independent variables. They are used in specialist applications such as the deformation of rubber-like materials where pressure is used as a primary variable.

**Hydrostatic stress: **Is the average of the direct stresses, ignoring the shear components. The hydrostatic stress is composed of principal stresses and causes a change of volume of an element but not its

shape.

**Hyperelastic: **Material behavior is used to describe materials, such as rubber and elastomer materials, which can sustain very large strains, e.g. over 400%, but still remain elastic.

**Ill-conditioning: **of matrices may occur if the round-off error become significant, i.e. the equation solvers becomes unstable. This can happen when rigid body motion takes place or when there are coefficients very different in magnitude in the same row, such as when a very high value of Young´s modulus is used for one material, whereas a very low one is used for another material.

**Implicit Euler method: **(also known as backward-difference) is a time-marching scheme in which the new value of the variable is arrived at using the tangent at the next time step. This method is unconditionally stable, i.e. it remains stable event if large time steps are used, although small time steps are more accurate.

**Integration/Gauss points: **Are the Gaussian integration points used to perform numerical integrations.

**Isoparametric elements: **Are elements in which the same shape functions are used to describe the geometry and the displacements.

**Isotropic hardening: **Occurs when the original yield surface increases in size with increasing plastic strain but maintains its original shape.

**Iterations: **Are an indirect way of solving non-linear problems, based on successive corrections of an initial guess (a trial solution)

**J-integral: **(also called J-contour integral) is a two dimensional path-independent energy integral used in fracture mechanics problems to calculate the intensity factor. The path starts from one surface of the crack, continues anti-clockwise inside the domain, and then ends on the other surface of the crack.

**Jacobian matrix: **Is a matrix used to transform a set of variables from one set of coordinate axes (such as Cartesian components) to another set of coordinate axes (such as local components).

**Kinematic hardening: **Occurs when the original yield surface is translated to a new position in the stress space as the plastic strain is increased, with no change in size or shape.

**Kinematically equivalent loads:**Are point loads that are applied to the nodes of an element to represent a distributed load. They are derived analytically to give the same work done as the distributed load., depending on the element shape functions.

**Langrarian elements: **Are elements that use Langrarian polynomials for the shape functions. They usually refer to elements in which the displacements are the only independent variables, such as continuum elements.

**Large deformations: **Usually occur in problems involving non-linear geometric behavior in which the deformations are so large that a new stiffness matrix has to be calculated at every iteration.

**Limit point: **Is a point on the load-displacement curve where the tangent is horizontal and the structure cannot sustain a further increase in load.

**Line search: **Is a method use in geometric non-linearity problems to accelerate the incremental-iterative procedure, but at the same time maintaining the reliability of the solutions by minimizing the total potential energy.

**Linear elastic fracture mechanics (LEFM): **Are those conditions which are applicable to a crack inside a loaded structure when the crack fields local to the crack tip are assumed to be elastic, and any plastic behavior is neglected.

**Load control: **Refers to problems where the load at a specified node or a set of nodes is increased by a small increment. Load control can be used in most applications, except where snap-through behavior is encountered, because the load must subsequently remain constant or decrease in order to follow the load-displacement curve.

**Local axes: **Are the axes used to describe the behavior of individual elements, without reference to the global axes. The Jacobian matrix is used to relate local variables to global variables. Local axes can also be used in prescribing boundary conditions and loads.

**Material non-linearity: **Occurs when the stress-strain constitutive relationships are non-linear as in plastic or creep behavior.

**Mesh: **Is usually used to refer to all the elements used to model a given problem.

**Mixed hardening: **Is a combination of isotropic and kinematic hardening where the original yield surface both expands and translates to a new position with increasing plastic strain.

**Mode shape: (**also know as eigenvector) Of a vibrating structure represents the deformed shape of the structure at a particular mode of vibration.

**Modes of fracture: **Exist as three separate deformation modes at any point along a crack profile, representing the basic effects of crack opening, shearing and tearing commonly known as modes I, II and III. In practice, combinations of these modes are usually present.

**Modify Newton-Raphson method: **Is a version of the Newton-Raphson method in which the tangent stiffness matrix is only updated at selected iterations.

**Natural frequency: **(also called resonant frequency or eigenvalue) is the frequency at which resonance occurs in a vibrating system.

**Newton-Raphson method: **Is a method of solving non-linear equations using an incremental-iterative approach based on calculating an updated tangent stiffness matrix, or gradient, for each iteration.

**Nodes or nodal points: **Are the points of the elements where the degrees of freedom (variables) are defined.

**Nominal strain: **(also called engineering strain) is defined as the change in length over the original length, and is suitable for elastic behavior when the change in length is relatively small.

**Nominal stress: **(also called engineering stress) is defined as the force divided over the original area. Nominal stress is appropriate for elastic behavior but is not suitable when the change in the cross-section area is large.

**Normality rule: **Is used in plasticity to ensure that the plastic strain components are in a ratio such that their resultant is in a direction normal to the plastic potential surface.

**Norton-Bailey equation: **Is a creep law in which the creep strain rate is proportional to stress, raised to some power, and time.

**Numerical integration: **Is the process of integrating the element stiffness matrix based on numerical algorithms such as Gaussian quadrature. Evaluations are made as strategic points within each element, usually at the Gauss points.

**Optimal points: **Are strategic locations within many element types where stress evaluations are especially accurate, often at the Gauss point locations.

**Patch tests:**Is a simple test of the potential performance of an element in which s state of constant strain is prescribed on a “patch” of elements.

**Perfectly plastic material: **Is a material where there is no further increase in the yield stress after initial yielding.

**Pin-jointed structures: **Are structures consisting of long straight line members connected by frictionless point-joints which cannot support any bending moment.

**Plane strain: **Is used to define very thick geometries where the strain across the thickness is neglected, but the stress there is non-zero.

**Plane stress:**Is used to define very thin geometries where the stress across the thickness is neglected.

**Plasticity: **Is a non-linear material behavior which describes the behavior of a metal once it is loaded beyond its yield point. The resulting plastic strains are permanent. Plasticity is usually assumed to be time independent.

**Plate elements:**Are used to model a plate under transverse loading which can sustain bending moments. Membrane action is not incorporated.

**Post-processor: **Is that part of an FE program used to examine the computed values, usually in interactive graphics plots, such as deformed shapes and stress contour plots.

**Potential energy: **Is the energy stored in a body due to the applied loads. This energy equals the stress energy less the potential of the applied forces, the product of each product component and the corresponding displacement.

**Pre-processor: **Is that part of an FE that deals with the generation of the data input, i.e. mesh generation, boundary conditions and load description.

**Primary creep: **Is the initial stage of creep where the strain rate decreases.

**Principal planes: **Are planes on which only the normal components of the stress remain and the shear components are zero. The stresses and strains acting on these planes are called principal stresses and principal strains.

**Principal stresses/strains: **Are normal stresses/strains with no shear components acting on the principal planes. The magnitudes of the principal stresses/strains are independent of the coordinate system used.

**Reduced integration: **Is a Gaussian integration scheme in which less integration points are used to integrate the stiffness matrix than theoretically required. However, it uses less computer time and has been shown to usually give greater accuracy to the final solution.

**Rheological models: **Are simplified models of springs (proportional to displacements) and dashpots (proportional to velocities) that can be used to represent viscoelastic and viscoplastic behavior.

**Rigid body motion: **Is the motion of the body in any direction that generates no stresses. Rigid body motion has to be prevented, otherwise the FE solutions become meaningless.

**Ritz method: **Is a technique used to solve partial differential equations by assuming a trial solution which satisfy the boundary conditions of the problems.

**Round-off errors: **Are small errors in numerical computations causes by terminating long numbers after a given number of significant digits.

**Secondary creep: **Is the creep stage where the creep strain rate is constant (or approximately constant)

**Shape functions: **Are the interpolation functions that describe both the geometry and the displacements. In isoparametric elements, the same shape functions are used to describe both shape and displacement.

**Shear locking: **Is a phenomenon which occurs when thick elements give overstiff results when modelling thin beams/plates/shells, due to an excess of shear energy being present. It can also affect 2D and 3D continuum elements.

**Shell elements: **Are used to model thin or thick shell structures in which membrane and bending actions are coupled.

**Snap-back: **Usually occurs where there is a vertical tangent in the load-displacement curve where the load on the body suddenly drops, but the displacement remains constant.

**Snap through: **Usually occurs where the deformed shape suddenly jumps from one position to another, but the load remains constant. Dynamic effects may be caused by the sudden movement. In the load-displacement path, snap through is exhibited by horizontal tangent in the load-displacement curve.

**Sparse matrix: **Is a matrix with the majority of its coefficients equal to zero such as the structural stiffness matrix.

**Stiffness matrix: **K, relates the nodal displacements to the nodal forces though the equation Ku=F. The stiffness matrix is symmetric and, when accumulated over the whole structure, is usually sparsely populated.

**Strain energy: **Is the energy gained in a body by the application of load, and equals the integrated product of every stress component and the corresponding strain component.

**Strain energy release rate: **Is, for a hypothetically small increase in crack length or area, the amount of strain energy released divided by that length or area.

**Strain hardening law: **Is used in analyzing creep behavior under a variable load where the creep strain rate is assumed to depend on the current stress and the accumulated creep strain.

**Strain hardening material: **Is a material where the stress after initial yielding increases with continuing plastic strains.

**Stress concentration factor: **Is the ratio of some important stress component at some location of high stress (such as at a corner or Arnold a small hole) divided by the corresponding value in areas of the same structure where the stresses are low.

**Stress intensity factor: **Is a fracture parameter at a crack tip when under conditions of LEFM. It is a function of applied load and crack length suitably dimensioned to have a finite value at the tip though the stresses are singular there, and may be used to characterize the state of fracture there.

**Stress relaxation: **Usually occurs in creep problems when the structure is loaded up to a stress level and then held at a constant strain.

**Substructures: **Also called super-elements, are mesh modeling techniques whereby a part of a structure, containing a number of elements, can be stored by the software as a single element. It can then be used for a variety of different purposes, just as if it were a new element type with its own stiffness matrix.

**Symmetric matrix: **Is a square matrix with its coefficients symmetrical about the main diagonal, i.e. ?ij = ?ji. The FE stiffness matrix is a symmetric matrix.

**Symmetry (of model): **Is a model reducing technique used in constructing a finite element mesh. It is applicable when the meshing of similar shapes with similar loading within the model can be avoided by using the principles of symmetry, and by using suitable boundary conditions. The different types of symmetry include repetitive, mirror, axial and cyclic. Asymmetric loading can also be modelled with suitable boundary conditions.

**Tangent stiffness matrix: **Is the updated stiffness matrix used in non-linear FE analysis the coefficients correspond to the derivatives of the residual forces with respect to the displacement degrees of freedom. This matrix is evaluated and used for each iteration solution in the Newton-Raphson method.

**Tertiary creep: **Is the creep stage where the strain rate increases very rapidly, followed by eventual failure.

**Time hardening law: **Is used in analyzing creep behavior under a variable load where the creep strain rate is assumed to depend on the current stress and the time from the start of the test

**Topology: **Of an element is the order in which the nodes of the element are defined.

**Transpose: **Of a matrix is obtained by interchanging its rows and columns.

**Tresca criterion: **Is a yield criterion which assumes that yielding stars when the maximum value of the shear stress reaches a given value determined from experiments.

**True strain: **(also called natural or logarithmic strain) is defined as the integral of the incremental change of length over the current length.

**True stress: **(also called Cauchy stress) is defined as the force over the current (instantaneous) area.

**Truss element: **Are used for pin-joined structures in which only axial forced are transmitted.

**Viscoelasticity: **Is a non-linear material behavior in which the behavior is time-dependent.

**von Mises stress/strain: **Is a scalar quantity defined to represent the individual stress or strain components at any point. It uses a formula involving the squares and differences of the individual stress/strain components.

**von Mises yield criterion: **Is a criterion of yielding which relates the multi-axial stresses in plasticity to the uniaxial behavior by using the concept of the critical value of the shear strain energy stored in the material.

**Weighted residual method: **Is a method of deriving the FE stiffness matrix by multiplying the partial differential equation by a suitable weighting function and minimizing the error.

**Yield criterion: **Is used to defined how the multi-axial behavior of the material is related to uniaxial behavior. Tresca and von Mises yield criteria.

**Yield function: **(or yield surface) Defines the values of stresses which will cause yielding both for initial yielding and also beyond yielding.

**Yield stress: **Is the stress value at which yielding occurs in a uniaxial test.

**Zero energy modes: **Are spurious element deformations that occur with zero strain energy, due to particular numerical integration schemes.