Hybrid-stress Finite Element Model Research Articles

Multiphase hybrid stress finite element analysis of heterogeneous media by simple mesh: One element with one interface

SummaryIn this paper, a finite element method is proposed to analyze the microscopic and macroscopic mechanical behaviors of heterogeneous media with randomly distributed inclusions. A simple mesh partitioned the domain into regular quadrilateral or triangular elements, where one element may contain two phases. An assumed stress hybrid formulation is implemented in the finite element model and the functional is derived for an element containing two phases. Numerical examples were used to study the microscopic and macroscopic properties of the composites, such as the effective modulus, to validate of the proposed model. The results show that the proposed multiphase hybrid stress finite element model can accurately measure the stress fields of materials with arbitrary microstructural distributions and improve computational efficiency by about 30 to 1500 times in comparison with the traditional displacement based finite element method.

A novel hybrid finite element analysis of bimaterial wedge problems

Abstract An assumed hybrid-stress finite element model together with a new super singular wedge-tip element with numerical eigensolutions is developed to study the bimaterial wedge/notch problems. The establishment of the super wedge-tip element consists of (1) a finite element method-based eigenanalysis is developed and applied to determine the order of the stress singularity and the angular dependences of the stress and displacement fields, (2) these fields are subsequently used to develop a finite element surrounding the wedge tip. To demonstrate the validity of the method, three types of the bimaterial wedge problems are examined and compared with analytical/referenced solutions. The high accuracy and general applicability of the present technique are shown.

Free vibration analysis of patched cracked composite laminates using a multilayer hybrid-stress finite element method

An assumed hybrid-stress finite element model incorporating two types of modified composite multilayer elements (MLTUP and MLTPH) is developed to investigate the free vibration of doubly and singly patched cracked laminates. Three-dimensional stresses of the laminated panel, patch and adhesive are accounted for. A composite patch material which has an efficient patching characteristic is employed. The effect of the patch parameters (length, thickness, width and position) on the fundamental natural frequency is also investigated. For a simply supported condition, the variation of the fundamental natural frequency after patching is negligible. However, the maximum variation of the fundamental natural frequency after patching for the problems solved under the cantilever edge condition is ca 10%. When the patch parameters are selected appropriately, the fundamental natural frequency after patching may become the same as that of the original uncracked laminate. This illustrates that the composite patch studied is a good choice for patching the cracked laminate tackled. The variations of the fundamental natural frequency for various crack lengths after patching are additionally presented. The technique developed can further be employed to treat a more realistic patching problem owing to that based on a three-dimensional viewpoint.

Multilayer hybrid-stress finite element analysis of composite laminates with delamination cracks originating from transverse cracking

An assumed hybrid-stress finite element model together with a composite multilayer element incorporating the technique of contact analysis is employed to analyze a composite laminate with delamination cracks originating from transverse cracking. Contact without friction on the crack surfaces and three-dimensional interlaminar stresses are taken into account. Once the contact zone exists, the assumed stress field satisfies the traction reciprocity conditions on the contact area and a transformation matrix is employed to deal with the contact conditions for the displacements on the contact zone. The interlaminar stresses and stress intensity factors are calculated accurately. As examples, cross-ply composite laminates with delamination cracks originating from transverse cracking under extension are studied. The influence of ply thickness on the growth of delamination cracks is also evaluated. The comparisons between the present results and referenced solutions show the high accuracy and validity of the present technique.

Estimation of stress intensity factors in partially patched cracked composite laminates by multilayer hybrid-stress finite element method

Abstract A multilayer hybrid-stress finite element model together with two types of composite multilayer elements (MLTUP and MLTPH) which have been developed to analyze patched crack is employed to evaluate the influence of various patch parameters (position, length, thickness and fiber orientation) on the estimation of stress intensity factors in partially patched cracked composite laminates. Three-dimensional stresses of the laminated panel, adhesive and patch are accounted for. The assumed stress field satisfies the equilibrium conditions within each layer, the interlaminar traction reciprocity conditions at interlaminar boundaries and traction-free boundary conditions on the bottom surface of the laminate. Once the traction boundary conditions on the top surface of the laminate are free, the conditions can be enforced by employing a Lagrangian multiplier technique. The interelement traction reciprocity condition at interelement boundaries is then satisfied through the introduction of other Lagrangian multipliers which are physically the interelement boundary displacements. The stress parameters assumed and the number of nodes taken are independent of the number of layers. The multilayer elements devised are therefore applicable to deal with the patched cracked laminate with a large number of layers. In addition, the technique developed also has the potentiality to treat general patched cracked laminates even with unknown stress singularities. The double and single patch configurations for both of the cracked (0° / 90°)s and ( −45° +45° ) s laminates under uniaxial tension are considered as examples. The patch made of composite material is a strip and covers only a part of the crack flanks. Excellent correlations between the present computed results with the referenced solutions show the versatility of the technique employed.

Multilayer hybrid-stress finite element analysis of a patched crack

Abstract An assumed hybrid-stress finite element model together with two types of composite multilayer elements (MLTUP and MLTPH) is developed to study the crack patching problem. Three-dimensional interlaminar stresses including transverse normal and shear stresses are accounted for. The assumed stress field satisfies: (1) the equilibrium conditions within each layer, (2) the interlaminar traction reciprocity condition at interlaminar boundaries and (3) traction-free boundary conditions on the top and bottom surfaces of the laminate. Since the stress parameters assumed and the number of nodes taken are independent of the number of layers, the multilayer elements devised are applicable to the laminate with a large number of layers. The singly and doubly patched cracked panels under uniaxial tension are studied. At the same time, with a bending field induced, the strength can be predicted accurately without using the bending correction factors. In addition, the maximum stress intensity factor is found at the unpatched free surface, and an unbalanced composite patch is devised to extend the safe life of the cracked panel. Excellent agreemenets between the present results and referenced solutions show the high accuracy and validity of the present technique.

Multilayer hybrid-stress finite element analysis of composite laminate with through-thickness cracks

Abstract An assumed hybrid-stress finite element model using a simple composite multilayer element is developed to analyze generally thin or moderately thick composite laminates with through-thickness cracks. The assumed stress field satisfies: (i) equilibrium conditions within each layer, (ii) the traction reciprocity conditions at interlaminar boundaries, and (iii) the traction-free boundary conditions at the top and bottom faces of the laminate. Since the number of nodes and assumed stress parameters are independent of the number of layers, the multilayer element devised is quite effective especially for the laminate with a large number of layers. Several selected composite laminates with a through-thickness edge crack are solved. Many fewer degrees of freedom and only a one-step solution are necessary for the present technique. The variations of mixed-mode stress intensity factors across the thickness of the composite laminate are also computed. Excellent agreements between the present results and referenced solutions are drawn. The technique developed is also applicable to analyze the structural behaviors of the cracked laminate with arbitrary fiber orientation and stacking sequence for which the stress singularity has not yet been found.

Buckling analysis of general composite laminates by hybrid-stress finite element method

An assumed hybrid-stress finite element model together with a composite multilayer element are developed to study the buckling of generally laminated composite plates with arbitrary thickness and edge condition under an in-plane stress system. the assumed stress field satisfies the equilibrium conditions within each layer, the interlaminar traction reciprocity conditions, and stress-free boundary conditions on the top and bottom surfaces of the laminate.

Three dimensional interlaminar stress analysis at free edges of composite laminate

Abstract A composite “multilayer element” based on an assumed hybrid-stress finite element model is presented. The stress field in each layer is derived from an assumed in-plane strain such that it satisfies equilibrium conditions within each layer and the interlaminar traction reciprocity conditions. The interelement boundary displacement is assumed to enforce the interelement traction reciprocity. The complex three dimensional stress analysis can thus be easily performed for realistic composite laminates, especially for the computation of the interlaminar stresses at straight and/or curved free edges. Excellent agreement between the present computed results and referenced solutions shows the high accuracy and efficiency of this work.

A three-dimensional hybrid stress isoparametric element for the analysis of laminated composite plates

Abstract In view of the increasing interest in using composite materials for aerospace structures, the analysis of laminated composite plates becomes essential. A three-dimensional eight-node hybrid stress finite element method is developed for the analysis of laminated plates. The hybrid stress model is based on the modified complementary energy principle and takes into account the transverse shear deformation effects. The displacement field is interpolated through shape functions and nodal displacements. All three displacement components are assumed to vary linearly through the thickness of each lamina. The stress field is interpolated through assumed stress polynomials with 55 stress parameters for each lamina. All six stresses are included and satisfy the homogeneous equilibrium equations. The validity of the hybrid stress finite element model is determined by comparing the predicted numerical results with the existing three-dimensional elasticity solutions. Excellent accuracy and fast convergence are observed in the numerical results.

Hybrid stress finite element analysis of bending of a plate with a through flaw

AbstractA hybrid stress finite element procedure for the solution of bending stress intensity factors of a plate with a through‐the‐thickness crack is presented. Reissner's sixth‐order plate theory including the effects of transverse shear deformation is used. The dominant singular crack tip stress field is embedded in the crack tip singular elements and only regular polynomial functions are assumed in the far field elements. The stress intensity factors can be calculated directly from the crack tip singular stress solution functions. The effects of the plate thickness, the ratio between the crack size and the inplane dimension of the plate, and the singular element size on the stress intensity factor solution are investigated.The effects of the explicit enforcement of traction‐free conditions along crack surfaces, which are the natural boundary conditions in the present hybrid stress finite element model, are also investigated. The numerical results of bending of a plate with a straight central crack compare favourably with analytical solutions. It is also found that the explicit enforcement of traction‐free conditions along crack surfaces is mandatory to obtain meaningful results for the Mode I type of bending stress intensity factor.

Fatigue crack propagation in titanium under general in-plane loading—II: Analysis

Numerical stress analysis employing the dislocation-superposition technique is carried out for curved crack configurations resulting from tensile fatigue loading of titanium Ti-6A1-4V plate containing an oblique center-crack. It is discovered that the tangential rate of growth of the fatigue crack along its path can be correlated with the range of the amplitude of the singularity in maximum principal stress at the crack tip. This finding is analogous to characterizations of crack growth rate vs stress-intensity range for conventional transverse-crack specimens. In addition, the calculated direction of the principal axis can be used to predict the general trend of the tangential direction of crack growth. These results demonstrate the significance of mixed-mode experiments in the study of mechanisms of fatigue crack propagation. A hybrid stress finite element model is used to analyze a typical crack configuration resulting from cyclic growth of an oblique center-crack, and the relative efficiencies of this and the dislocation-superposition method for such calculations are discussed.

Hybrid stress finite element model for non‐linear shell problems

AbstractThe present paper describes a hybrid stress finite element formulation for geometrically non‐linear analysis of thin shell structures. The element properties are derived from an incremental form of Hellinger‐Reissner's variational principle in which all quantities are referred to the current configuration of the shell. From this multi‐field variational principle, a hybrid stress finite element model is derived using standard matrix notation. Very simple flat triangular and quadrilateral elements are employed in the present study. The resulting non‐linear equations are solved by applying the load in finite increments and restoring equilibrium by Newton‐Raphson iteratioin. Numerical examples presented in the paper include complete snap‐through buckling of cylindrical and spherical shells. It turns out that the present procedure is computationally efficient and accurate for non‐linear shell problems of high complexity.

On the hybrid stress finite element model for incremental analysis of large deflection problems

An extension of the hybrid stress finite element model, originally suggested for linear small displacement problems by Pian, to the large deflection problem is presented. An incremental approach and the concepts of initial stress are employed. A procedure to check the equilibrium in the reference state, prior to the addition of a further load increment, is included. An example and a discussion of the results are presented. Дaeтcя oбoбщeниe мoдeли кoнeчнoгo злeмeнтa co cмeщaнными иaпpяжeниями для зaдaчи c бoльщими пpoгибaми. Этa мoлeль пepвoнaчaльнo пpeдлoжeнa Пиaнoм для линeиныч зaдaч, в paмкaч мaлыч пepeмeщeний. Иcпoльзyютcя paзнocтный пoдчoд и пoнятня нaчaльныч нaпpяжeний. Пpивoдитcя пpoцecc пpoвepки paвнoвecия для иcчoднoгo cocтoяния, paньщe дoбaвлeния дoбaвoчнoгo пpиpaщeния нaгpyзки. Дaютcя пpимep и oбcyждeниe peзyльтaтoв.

Bounds to the influence coefficients by the assumed stress method

Abstract The direct influence coefficient obtained by the hybrid stress finite element model proposed by Pian has been proved to be bounded from above from that of the equilibrium model, provided that they have the same type of stress distribution within each element, and that of the compatible displacement model, provided that they are the same type of displacement along the interelement boundary.