Material Stiffness Matrix Research Articles

On the symbolic manipulation and code generation for elasto-plastic material matrices

A computerized procedure for symbolic manipulations and FORTRAN code generation of elasto-plastic material matrix for finite element applications is presented. Special emphasis is placed on expression simplifications during intermediate derivations, optimal code generation, and interface with the main program. A systematic procedure is outlined to avoid redundant algebraic manipulations. Symbolic expressions of the derived material stiffness matrix are automatically converted to RATFOR code which is then translated into FORTRAN statements through a preprocessor. To minimize the interface problem with the main program, a template file is prepared so that the translated FORTRAN statements can be merged into the file to form a subroutine (or a submodule). Three constitutive models; namely, von Mises plasticity, the Drucker-Prager model, and a concrete plasticity model, are used as illustrative examples.

Nonlinear analysis of anisotropic panels

A computational procedure is presented for the efficient nonlinear analysis of symmetric anisotropic panels. The three key elements of the procedure are: 1) use of three-field mixed models having independent interpolation (shape) functions for stress resultants, strain components, and generalized displacements, with the stress resultants and strain components allowed to be discontinuous at interelement boundaries; 2) decomposition of the material stiffness matrix into the sum of an orthotropic and a nonorthotropic (anisotropic) part; and 3) successive application of the finite element method and the classical Rayleigh-Ritz technique. The finite element method is first used to generate a few global approximation vectors (or modes). Then the amplitudes of these modes are computed by using the Rayleigh-Ritz technique. The global approximation vectors are taken to be various-order derivatives of the strain components, stress resultants, and generalized displacements with respect to an anisotropic tracing parameter and a load parameter; they are evaluated at zero values of the two parameters. The size of the analysis model used in generating the global approximation vectors is identical to that of the corresponding orthotropic structure. The effectiveness of the proposed computational procedure is demonstrated by means of a numerical example, and its potential for solving quasi-symmetric nonlinear problems of composite structures is discussed.

A large strain finite element formulation for one dimensional membrane elements

A finite element formulation suitable for large strain analysis of one-dimensional membranes is described which has applications to the analysis of problems in soil mechanics involving reinforced earth. The finite element equations, which are derived for a plane strain iso-parametric element of arbitrary order, are cast in the familiar small displacement form, but with a modified material stiffness matrix. In this formulation, full account is taken of element rotations during each load step, an effect which is important when stresses cease to be insignificant when compared with the material modulus. The method is illustrated with reference to a large displacement test problem which has a known closed form solution. An example is included demonstrating the application of the formulation to the analysis of a reinforced unpaved road.

Incrementalization procedure for elasto‐plastic constitutive model with multiple, intersecting yield surfaces

AbstractConstitutive models for description of the non‐linear stress‐strain behaviour of engineering materials are often developed and presented in forms which cannot be employed directly in non‐linear matrix computer analysis codes. Presented herein is a procedure for developing the incremental stiffness matrix for an elasto‐plastic material model with multiple, intersecting yield surfaces. Associated or non‐associated flow may occur on the individual yield surfaces. The material stiffness matrix is non‐symmetric if non‐associated flow occurs on any one of the yield surfaces. In order to illustrate the matrix forms which are generated, the case of a recently developed model with two yield surfaces is presented in detail. This model involves associated flow on one and non‐associated flow on the other yield surface.

Approximation of the plastic buckling load as the solution of an eigenvalue problem

Plastic buckling analysis of thin shell structures is generally performed by employing a Newton's type method within an incremental approach. This process becomes more efficient if a good estimate of the critical buckling load is available. This one can be obtained as the solution of an eigenvalue problem that generalizes the elastic bifurcation analysis. The stability matrix of the eigenvalue problem involves an additional term with respect to the elastic bifurcation problem; that is the material stiffness matrix that accounts for the change of the tangent modulus along the fundamental path. The proposed approach is applied to the elastoplastic buckling of spherical caps under pressure.

Behavior of Concrete under Biaxial Stresses

The results of an extensive series of tests of three types of concrete under biaxial loadings are used to develop stress-strain relations for concrete subjected to biaxial stress states. By means of a decomposition of the stresses and strains into their hydrostatic and deviatoric portions, it was possible to express the relations between octahedral normal stresses and strains, and octahedral shear stresses and strain through use of bulk and shear moduli. These moduli can be expressed as functions of the octahedral shear stress only; formulas and coefficients are given for the values of the tangent and secant, bulk and shear moduli for the three types of concrete. The deformational behavior is described as the material reaches its failure stage. The application of these nonlinear stress-strain relations to stress analysis is indicated; a material stiffness matrix for use in finite element analysis is presented, and a partial differential equation with variable coefficients for analysis of plane-stress problems is shown.

Computer simulation of extended defects in metals

A novel nonlocal lattice particle framework is proposed to investigate the microstructural effects, such as the crystallographic orientation distribution and grain boundary properties, on the mechanical performance of 2D polycrystalline materials. The classical approach of treating material anisotropy in other numerical methods, such as finite element method, is by transforming the material stiffness matrix for each crystallite. In the proposed method, the polycrystalline microstructures are constructed by rotating the underlying topological lattice structure consistently with the material crystallographic orientation while keeping the material stiffness matrix intact. By rotating the underlying lattice structure, the grain boundaries between different grains are naturally generated at locations where two crystallites meet. Thus, the grain boundary effect on the performance of the crystalline aggregates can be naturally incorporated. Parametric studies on the effects of crystallographic orientation distribution on both elastic and fracture behavior of polycrystalline materials are performed. The simulation results are compared with both analytical solutions and experimental observations in the open literature. Conclusions and discussions are drawn based on the current study.