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.
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.
Used in fracture mechanics to evaluate fracture parameters at a single crack tip calculated using an expression integrated along a path surrounding the tip.
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.
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.
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.
A method for introducing constraints into an analysis where the effects of the constraint are represented in terms of the unknown Lagrange multiplying factors.
A geometrically non-linear formulation where the equilibrium conditions are satisfied in the fixed reference configuration. See also Eulerian Formulation.
For non-linear large deflection problems the equations can be defined in various ways. 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. See also Eulerian Method.
A method for finding the first few Eigenvalues and eigenvectors of a set of equations. It is very well suited to the form of equations generated by the finite element method. It is closely related to the method of conjugate gradients used for solving simultaneous equations iteratively.
Lay up of individual plies or layers to form laminated material. Plies may be arranged in alternating fibre orientations to produce favourable multidirectional strength.
Minimisation of the sum of the squares of the distances between a set of sample points and a smooth surface. The finite element method gives a solution that is a least squares fit to the equilibrium equations.
Points at which the tangent to the load-displacement curve becomes either horizontal or vertical and the structural stiffness matrix becomes singular under load or displacement control, respectively.
A technique for modelling part-through cracks in shell type structures. An equivalent distributed spring replaces the crack with matching compliance, so the curvature is effectively ignored but the modelling is easier.
One or more rows (columns) of a matrix are linear combinations of the other rows (columns). This means that the matrix is singular. See also Singular Matrix.
A given crack inside a loaded structure behaves in conditions of LEFM if the crack fields local to the crack tip are assumed to be elastic, and any plastic behaviour is neglected.
When the coefficients of stiffness, mass and damping are all constant then the system is linear. Superposition can be used to solve the response equation.
A means of advancing a non-linear solution using a load parameter: this is the conventional method, others being displacement and theArc Length Method.
For softening materials, a tendency for non-linear behaviour to concentrate into local bands, requiring special treatment. See also Non-Linear Analysis.
The assumed displacement form of the finite element solution gives a lower bound on the maximum displacements and strain energy (i.e. these are under estimated) for a given set of forces. This is the usual form of the finite element method. See also Strain Energy.
The constant(s) of proportionality relating the acceleration(s) to the force(s). For a discrete parameter multi-degree of freedom model, this is usually given as a mass matrix.
The matrix relating acceleration to forces in a dynamic analysis. This can often be approximated as a diagonal matrix with no significant loss of accuracy.
An information system used to manage physical testing of engineering materials and the derivation, management and dissemination of material performance data.
The material flexibility matrix is the inverse of the Material Stiffness Matrix allowing the strains to be found from a given set of stresses. The resulting matrix must be symmetric and positive definite.
The physical properties required to define the material behaviour for analysis purposes. For stress analysis typical required material properties are Young's modulus, Poisson's ratio, density and coefficient of linear expansion. The material properties must have been obtained by experiment.
The material stiffness matrix allows the stresses to be found from a given set of strains at a point. This matrix must be symmetric and positive definite. See also Material Flexibility Matrix.
If matrix A times matrix B gives the unit matrix then A is the inverse of B (B is the inverse of A). A matrix has no inverse if it is singular.See also Singular Matrix.
A form of notation for writing sets of equations in a compact manner. Matrix notation highlights the generality of various classes of problem formulation and solution. See also Matrix Algebra.
Membrane behaviour is where the strains are constant from the centre line of a beam or centre surface of a plate or shell. Plane sections are assumed to remain plane. A membrane line element only has stiffness along the line, it has zero stiffness normal to the line. A membrane plate has zero stiffness normal to the plate. This can cause zero energy (no force required) displacements in these normal directions. If the stresses vary linearly along the normal to the centre line then this is called bending behaviour.
The progressive refinement of element size and positioning in mesh models (H-refinement) or increase in order of element type (p-refinement) to produce improvements in solution accuracy.
The mesh density indicates the size of the elements in relation to the size of the body being analysed. The mesh density need not be uniform all over the body There can be areas of mesh refinement (more dense meshes) in some parts of the body. See also Mesh Refinement.
The creation of a suitable mesh model, to represent the given structure with suitable refinement in regions of high field variation, good representation of boundaries, and incorporating all other required features.
The process of generating a mesh of elements over the structure. This is normally done automatically or semi-automatically. This is also referred to as Element Generation.
There can be areas of mesh refinement (more dense meshes) in some parts of the body. Making the mesh finer is generally referred to as h-refinement. Making the element order higher is referred to as P-Refinement.
The damping associated with the generalised displacements defined by the eigenvectors. Its value has no physical significance since the eigenvector contains an arbitrary normalising factor.eigenvectors contain an arbitrary normalising factors.
The mass associated with the generalised displacements defined by the eigenvectors. Its value has no physical significance since the eigenvector contains an arbitrary normalising factor but the ratio of modal stiffness to modal mass is always the eigenvalue.
The stiffness associated with the generalised displacements defined by the eigenvectors. Its value has no physical significance since the eigenvectors contain arbitrary normalising factors but the ratio of modal stiffness to modal mass is always the Eigenvalues.
A systems engineering methodology that focuses on creating and exploiting domain models as the primary means of information exchange between engineers, rather than on document-based information exchange.
Three separate deformation modes exist 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.
A Newton-Raphson solution in which the tangent stiffness matrix is updated only at the beginning of every increment. See also Newton-Raphson Non-Linear Solution.
A generalisation of the Coulomb friction failure law; used for concrete, rock and soils, where the hydrostatic stress does not influence yielding. The criterion is an inverted hexagonal pyramid in principal stress space.
Used in CFD to accelerate the convergence of iterative solution techniques based on the solution of a set of simultaneous correction equations, and allowing a reduction in the number of equations to be solved.
