B. D. Reddy

A new locking-free brick element technique for large deformation problems in elasticity

Abstract In the present contribution, an innovative brick element formulation for large deformation problems in finite elasticity is discussed. The new formulation can be considered as a reduced integration plus stabilization concept with the stabilization factors being computed on the basis of the enhanced strain method. Such an idea has not been applied yet in the context of large deformation 3D problems and leads to a surprisingly well-behaved locking-free element formulation. Crucial to the method is the notion of the so-called equivalent parallelepiped. The major advantages of this element technology are its simplicity and robustness. Since the element quantities are evaluated only in the center of the element, the approach is also very efficient from the numerical point of view.

An experimental study of the plastic buckling of circular cylinders in pure bending

Abstract An experimental investigation into the plastic buckling of cylindrical tubes subjected to bending moments at the ends is reported on. Suitable parameters by means of which the buckling moment may be represented are first discussed, and after a description of the apparatus and the testing procedure, the results of tests on stainless steel and aluminium alloy tubes are given. These results are compared with analytical results for the collapse of cylinders under pure bending, and uniform axial compression. The mode of deformation of the cylinders is discussed and the experimental strains are compared with those of others for tests on axially compressed cylinders as well as cylinders in pure bending. The strains lie within ± 30% of those predicted by J 2 deformation theory for cylinders in axial compression: the corresponding range of stresses is about ± 5%.

Stability and convergence of a class of enhanced strain methods

A stability and convergence analysis is presented of a recently proposed variational formulation and finite element method for elasticity, which incorporates an enhanced strain field. The analysis is carried out for problems posed on polygonal domains in

A new stabilization technique for finite elements in non‐linear elasticity

R^n

Algorithm for the solution of internal variable problems in plasticity

, the finite element meshes of which are generated by affine maps from a master element. The formulation incorporates as a special case the classical method of incompatible modes. The problem initially has three variables, viz, displacement, stress, and enhanced strain, but the stress is later eliminated by imposing a condition of orthogonality with respect to the enhanced strains. Two other conditions on the choice of finite element spaces ensure that the approximations are stable and convergent. Some features of nearly incompressible and incompressible problems are also investigated. For these cases it is possible to argue that locking will not occur, and that the only spurious pressures present are the so-called checkerboard modes. It is shown th...

Aortic valve leaflet mechanical properties facilitate diastolic valve function

In the present contribution, an innovative stabilization technique for two-dimensional low-order finite elements is presented. The new approach results in an element formulation that is much simpler than the recently proposed enhanced strain element formulation, yet which gives results of at least the same quality. An important feature in the regime of large deformations is the stability of the element, which is addressed in detail. The main advantages of the new formulation are, besides its simplicity, its computational efficiency and robust behaviour. Only three history variables have to be stored, making this stabilization concept particularly interesting for large-scale problems. Copyright

Applications of mathematical programming concepts to incremental elastic-plastic analysis☆

A variational formulation of the quasistatic boundary value problem of plasticity is presented. An internal variable formulation is used, and the evolution or flow law is expressed in terms of the dissipation function; this form is a dual representation of the more conventional form, in which the flow law is written in terms of the yield function. Finite dimensional approximations of the variational problem are derived, and algorithms for the solution of such approximations are discussed. A generalised midpoint rule approximation in time is used, and algorithms of predictor-corrector type are presented and discussed; we show that the consistent moduli arising in the algorithms are symmetric. Finally, a stability analysis is carried out; it is shown that the algorithm is nonlinearly stable for 12 ≤ α ≤ 1 where α is the midpoint rule parameter. The measure of stability used is B-stability, a measure exploited in a recent study of return mapping algorithms by Simo and Govindjee.

Mixed finite element methods for the circular arch problem

This work was concerned with the numerical simulation of the behaviour of aortic valves whose material can be modelled as non-linear elastic anisotropic. Linear elastic models for the valve leaflets with parameters used in previous studies were compared with hyperelastic models, incorporating leaflet anisotropy with pronounced stiffness in the circumferential direction through a transverse isotropic model. The parameters for the hyperelastic models were obtained from fits to results of orthogonal uniaxial tensile tests on porcine aortic valve leaflets. The computational results indicated the significant impact of transverse isotropy and hyperelastic effects on leaflet mechanics; in particular, increased coaptation with peak values of stress and strain in the elastic limit. The alignment of maximum principal stresses in all models follows approximately the coarse collagen fibre distribution found in aortic valve leaflets. The non-linear elastic leaflets also demonstrated more evenly distributed stress and strain which appears relevant to long-term scaffold stability and mechanotransduction.

Mechanics of cranial sutures using the finite element method

Abstract As a conceptual framework for studying problems in elastic-plastic structural mechanics, internal variable and mathematical programming formulations have provided valuable insights. This paper attempts to extend these insights. It is argued, from an internal variable formulation, that the appropriate method of analysis of elastic-plastic problems is by means of an incremental holonomic constitutive equation which relates total stresses to internal variable changes. The relationship between Newton-Raphson type iteration schemes and the solution of the programming problem associated with the incremental holonomic problem is also explored.

Uniform convergence and a posteriori error estimators for the enhanced strain finite element method

Abstract The boundary-value problem for linear elastic circular arches is studied. The governing equations are based on the Timoshenko-Mindlin-Reissner assumptions. The problem is formulated in both the standard and mixed variational forms, and after establishing the existence and uniqueness of solutions to these equivalent problems, the corresponding discrete problems are studied. Finite element approximations of the mixed are shown to be stable and convergent, and selective reduced integration applied to the standard discrete problem renders it equivalent to the mixed problem. The results of numerical experiments are presented; these confirm the convergent behaviour of the mixed problem. For the standard problem with full integration, convergence is suboptimal or nonexistent for small values of the thickness parameter, while for the mixed or reduced integration problem, the numerical rates of convergence coincide with those predicted by the theory.