Membrane Strain Fields Research Articles

Pneumatic Non-Equibiaxial Cell Stretching Device With Live-Cell Imaging.

Adherent cell behavior is influenced by a complex interplay of factors, including chemical and mechanical signals. In vitro experiments that mimic the mechanical environment experienced by cells in vivo are crucial for understanding cellular behavior and the progression of disease. In this study, we developed and validated a low-cost pneumatically-controlled cell stretcher with independent control of strain in two directions of a membrane, enabling unequal biaxial stretching and real-time microscopy during actuation. The stretching was achieved by two independent pneumatic channels controlled by electrical signals. We used finite element simulations to compute the membrane's strain field and particle tracking algorithms based on image processing techniques to validate the strain fields and measure the cell orientation and morphology. The device can supply uniaxial, equibiaxial, and unequal biaxial stretching up to [Formula: see text] strain in each direction at a frequency of [Formula: see text], with a strain measurement error of less than 1%. Through live cell imaging, we determined that distinct stretching patterns elicited differing responses and alterations in cell orientation and morphology, particularly in terms of cell length and area. The device successfully provides a large, uniform, and variable strain field for cell experiments, while also enabling real-time, live cell imaging. This scalable, low-cost platform provides mechanical stimulation to cell cultures by independently controlling strains in two directions. This could contribute to a deeper understanding of cellular response to bio-realistic strains and could be useful for future in vitro drug testing platforms.

A corotational isogeometric assumed natural strain shell element in updated lagrangian formulation for general geometric nonlinear analysis of thin-walled structures

A new general geometric nonlinear curved quadrilateral shell element, which can handle large displacements and large rotations, is presented in this paper. It is formulated based on the incremental virtual work equation in updated Lagrangian formulation for general large displacement analysis. The kinematics of shell element is described using non-uniform B-spline patch, which results in the element displacement field with higher order continuity and the element performance more resistive to shear locking and membrane locking. In order to perfectly avoid such membrane locking and shear locking behaviors, the assumed in-plane membrane strain field and the assumed transverse shear strain field for the incremental covariant Green-Lagrange strain are constructed as linear combinations of non-uniform B-spline basis functions with degrees lower than those adopted by the original displacement field interpolation. In addition, based on Euler parameters, the corotational procedure available for high order isogeometric shell element is developed to extract the strain-producing deformational displacements and rotations and to update the element stresses and realistic internal force vectors. The accuracy and robustness of proposed shell element are demonstrated through the solutions of various benchmark problems.

The simplified MITC4+ shell element and its performance in linear and nonlinear analysis

The recently developed improved MITC4+ shell element exhibits excellent convergence behavior in both bending- and membrane-dominated shell problems. Compared to the widely recognized MITC4 shell element, the membrane and in-plane shear locking were significantly alleviated. However, it is difficult to implement the assumed membrane strain field because of its complexity. In this study, a simplified MITC4+ shell element is developed. A simpler assumed membrane strain field that facilitates the extension to nonlinear formulations is proposed. To reduce the sensitivity of the element performance to mesh distortion, a geometry-dependent Gauss integration scheme is employed. The proposed element passes the patch, zero-energy mode, and isotropy tests. The performance of the simplified MITC4+ shell element is demonstrated using various linear and nonlinear benchmark problems.

The strain-smoothed MITC3+ shell element in nonlinear analysis

This paper presents the strain-smoothed MITC3+ shell finite element for nonlinear analysis. The strain-smoothed MITC3+ shell element, recently developed for linear analysis, significantly improves the membrane behavior of the MITC3+ shell element while retaining its excellent bending behavior. The achievement is attained by adopting the strain-smoothed element (SSE) method to the membrane strain field. To extend its formulation to nonlinear analysis, the total Lagrangian formulation is employed allowing for large displacements and rotations. Through various numerical examples, we verify that the strain-smoothed MITC3+ shell element also shows the superior performance in geometrical and material nonlinear analyses.

Isogeometric MITC shell

This paper proposes an isogeometric formulation of the Reissner–Mindlin shell, using the Mixed Interpolation of Tensorial Components (MITC) technique to alleviate shear locking and membrane locking. Instead of over each Non-Uniform Rational B-Spline (NURBS) element, kinematics of the shell is directly formulated on the entire NURBS patch, with its displacements decoupled to the motions of control points and of director vectors tied to the control points. Since control points are in general not located on the surface, those control-point director vectors are interpolated from normal vectors at a set of pre-defined integration points on the patch. The assumed in-plane membrane strain field and the assumed transverse shear strain field are built as linear combinations of NURBS basis functions with degrees lower than those employed by the displacement interpolation, and tied to the original covariant strain fields at a set of well-selected tying points. Integration points for the definition of control-point director vectors and tying points of the assumed covariant strain fields are provided by the optimal and reduced macro-element quadrature rules, respectively. In this way, the classical MITC technique is fully incorporated into the isogeometric framework with arbitrary polynomial orders or patch configurations. Thanks to the high smoothness of NURBS basis functions, the geometry-preserving nature of isogeometric transformation, and the locking-resistive capability of MITC technique, the isogeometric MITC shell formulation shows superior convergence behavior, minimal geometrical error, and good robustness against patch distortion. In particular, its convergence property is insensitive to NURBS polynomial order or control point density. These advantages are demonstrated through a number of well-established benchmark problems, validating the proposed isogeometric MITC formulation being an accurate and efficient solution for linear analysis of shell structures.

The strain-smoothed MITC3+ shell finite element

In this paper, we propose the strain-smoothed MITC3+ shell finite element. The membrane strain field of the continuum mechanics based 3-node triangular shell element (MITC3+) is smoothed using the recently developed strain-smoothed element (SSE) method. The strain-smoothed MITC3+ shell element passes basic tests (patch, isotropy and zero energy mode tests) and shows significantly improved membrane behavior in various numerical examples. The major advantage of the SSE method is that no additional degree of freedom is required for solution improvement.

Polygonal shell elements with assumed transverse shear and membrane strains

In this study, a new polygonal shell element is developed to provide greater flexibility in mesh design of complex shell structures. Wachspress coordinates are used to construct shape functions for polygonal shell elements. An assumed covariant shear strain field with respect to the element natural coordinate system is defined by employing the mixed interpolation of tensorial components approach to avoid transverse shear locking in polygonal shell elements. Moreover, an assumed covariant membrane strain field is constructed by using characteristic geometry and displacement vectors defined on quadrilateral subdomains of polygonal shell elements to alleviate membrane locking due to the element curvature. Some benchmark shell problems are solved to evaluate the performance of the proposed polygonal shell elements. Numerical experiments show that they converge much better than triangular shell elements and comparable to quadrilateral shell elements.

On the Molecular Filters in Cochlea Transduction

The article is devoted to the specific consideration of the cochlear transduction for the low level sound intensities, which correspond to the regions near the perception threshold. The basic cochlea mechanics is extended by the new concept of the molecular filters, which allows discussing the transduction mechanism on the molecular level in the space-time domain. The molecular filters are supposed to be built on the set of the stereocilia of every inner hair cell. It is hypothesized that the molecular filters are the sensors in the feedback loop, which includes also outer hair cells along with the tectorial membrane and uses the zero compensation method to evaluate the traveling wave shape on the basilar membrane. Besides the compensation, the feedback loop, being spatially distributed along the cochlea, takes control over the tectorial membrane strain field generated by the outer hair cells, and implements it as the mechanism for the automatic gain control in the sound transduction.

An interface shell element for coupling non-matching quadrilateral shell meshes

In this study, a novel interface shell element (ISE) is developed based on a variable-node element formulation to couple non-matching quadrilateral shell meshes. Shape functions for ISEs are explicitly presented in a polynomial form with the use of appropriate supports of weight functions in moving least square (MLS) approximation. Assumed natural strains in the form of the mixed interpolation of tensorial components (MITC) approach are employed to avoid the transverse shear locking when the thickness of shell tends to zero. Moreover, an assumed membrane strain field defined over quadrilateral subdomains subdividing an ISE is used to alleviate the membrane locking in curved ISEs. Numerical experiments show the effectiveness and efficiency of ISE for connecting dissimilar quadrilateral shell meshes at a common interface.

A reconstructed local [formula omitted] formulation for isogeometric Kirchhoff–Love shells

An efficient assumed strain formulation for avoiding the membrane locking of non polar shells is developed in the context of B-spline interpolation. Assumed membrane strains are introduced locally in each element with a local L2-projection, and then a spline reconstruction algorithm, developed by Thomas et al. (2015), is employed for reconstructing the membrane strain at the global patch level. In this way, a banded stiffness matrix is obtained that strongly reduces the computational cost with respect to the standard B̄ formulation for isogeometric analysis. The main advantages of the proposed method are: the reproduction of regular membrane strain fields, an accurate membrane locking-free solution and an high computational efficiency with respect to standard B̄ formulation. The method can be applied for any polynomial degree of the B-spline interpolation. The effectiveness of the proposed method is demonstrated analyzing some bending and membrane dominated problem, commonly employed as benchmark in the shell literature.

A new MITC4+ shell element

We presented in Ko et al. (2016) an MITC4+ shell element that shows a much improved convergence behavior when compared to the widely used MITC4 (Dvorkin and Bathe, 1984) shell element. However, the element does still not show optimal convergence behavior when used in distorted meshes in the analysis of some shell problems. In the present paper, we establish a new MITC4+ shell element which shows significantly improved and indeed an almost optimal convergence behavior. In this new continuum mechanics-based shell element, the shear locking is alleviated using the well-known assumed transverse shear strain field of the original MITC4 shell element and the membrane locking is alleviated using a new assumed membrane strain field. The new MITC4+ shell element passes all basic tests: the isotropy, zero energy mode and patch tests. The excellent performance of the shell element is demonstrated through convergence studies in the solution of well-selected behavior-encompassing shell benchmark problems.

A microscale anisotropic biaxial cell stretching device for applications in mechanobiology

A multi-layered polydimethylsiloxane microfluidic device with an integrated suspended membrane has been fabricated that allows dynamic and multi-axial mechanical deformation and simultaneous live-cell microscopy imaging. The transparent membrane’s strain field can be controlled independently along two orthogonal directions. Human foreskin fibroblasts were immobilized on the membrane’s surface and stretched along two orthogonal directions sequentially while performing live-cell imaging. Cyclic deformation of the cells induced a reversible reorientation perpendicular to the direction of the applied strain. Cells remained viable in the microdevice for several days. As opposed to existing microfluidic or macroscale stretching devices, this device can impose changing, anisotropic and time-varying strain fields in order to more closely mimic the complexities of strains occurring in vivo.Electronic supplementary materialThe online version of this article (doi:10.1007/s10529-013-1381-5) contains supplementary material, which is available to authorized users.

Physically stabilised displacement-based ANS solid-shell element

A robust reduced integration solid-shell finite element in irreducible variational formulation is demonstrated in this paper. To enable shell element-like performance using a single through-shell-thickness element layer, we apply element technologies isotropically, introducing local tri-quadratic hierarchical displacement modes thereafter systematically projected to the global coordinate frame. We also enforce uniform, bilinear and linear ANS interpolations to all local normal and shear strains respectively, imposing identical in-plane and out-of-plane element response for robust 3D thin shell deformation behaviour in non-regular, trapezoidal-shaped meshes. The hierarchical strains realised improve the element performance by complementing the locking alleviating actions of the ANS-modified membrane and shear strain fields in reduced numerical integration scheme performed along the designated shell thickness. We restrict our initial numerical investigations to geometrically linear beams, plates and shell problems solely for preliminary assessment of the element's in-plane and out-of-plane behaviour in response to the uniform modification of strains and hierarchical displacement modes introduced. To this end, the results indicate robust behaviour and comparable accuracy in comparison to mixed variational, enhanced assumed strain elements in full and in reduced integration in all structural problem classes undertaken.

A mixed co-rotational curved quadrilateral shell element

This paper presents a 9-node isoparametric formulation of degenerated curved shell element, which is based on a co-rotational framework. Within this framework, the local reference axes are easily defined by the two diagonal vectors generated by the four corner nodes in the current deformed configuration. Furthermore, vectorial rotational variables are utilised, which consist of the two smaller components of each nodal orientation vector, enabling additive and commutative rotational operations. The element strain energy is evaluated in the local co-rotational system using the Green strain, where the Hellinger-Reissner functional is employed with assumed strain tensors for the membrane and shear strain fields to alleviate the membrane/shear locking phenomena. By defining the equilibrium conditions in terms of the out-of-balance between the work-conjugate internal and external forces, symmetric element tangent stiffness matrices are achieved in both the local and global systems, thus leading to considerable computational benefits. Finally, several examples of elastic shells subject to large displacements are considered, which demonstrate favourable performance of the proposed finite element formulation.

Development of microstretching device to evaluate cell membrane strain field around sensing point of mechanical stimuli

In this study, we developed a novel cell-stretching device to evaluate the micro-strain field on a cell membrane in which a cellular response to a mechanical stimulus occurred. In a conventional cell-stretching device, rigid displacement of the cell body during the stretching operation is not negligible. This rigid displacement makes it difficult to observe the initial cellular response to stretching with high spatial and temporal resolution. A novel device is fabricated by applying the MEMS (micro-electro-mechanical systems) technique, and consists of a transparent elastic microchamber and a microlinkage mechanism by which uniaxial stretch is applied to the microchamber. The microchamber and linkage mechanism are fabricated by micromolding and multiple exposure of photoresist, respectively. We demonstrated the applicability of the fabrication process that we proposed, and evaluated the performance of the microdevice prototype. As a result of operation tests, rigid displacement of the cell body during stretching was found to be well suppressed. This result indicated the potential of this device to be used to observe and evaluate the initial cellular response and micro-strain field on a cell membrane during uniaxial stretching.

One point quadrature shell elements: a study on convergence and patch tests

One point quadrature shell elements are being widely used in the numerical simulation of shell structures, including sheet forming, because essentially of their computational efficiency. Nowadays, the purpose of using one point quadrature shell elements is not only related to computational efficiency but also because these elements have shown to be simultaneously robust and accurate in the simulation of complex sheet metal forming processes. The main objective of this work is to study the convergence behavior of different one-point quadrature shell elements and their ability to pass the membrane and bending patch tests. For comparison purposes, two new elements include a new formulation for the membrane strain field in order to further improve the membrane behavior of the element developed in previous work of (in Cardoso et al. Comput Meth Appl Mech Eng 191:5177, 2002). The original convective membrane strains of Cardoso et al. (Comput Meth Appl Mech Eng 191:5177, 2002) (in the stabilization matrices only) are thus replaced by new membrane strains, constructed directly at the co-rotational coordinate system (located at the element’s center). It is thus proved that with this new membrane formulation the elements pass now all the patch tests but, for warped (or curved) element geometries, their accuracy is not as good as the original element of (Cardoso et al. in Comput Meth Appl Mech Eng 191:5177, 2002) based on the convective coordinate system. In the numerical results presented in this paper, comprehensive comparison and discussion of these formulations are made for well known linear benchmark examples.

Elastodynamic analysis of low tension cables using a new curved beam element

In this paper, we address and overcome the difficulties associated with the use of the classic cable theory to treat low tension cables by developing a new three-noded locking-free nonlinear curved beam element. Based upon nonlinear generalized curved beam theory, large deformations and rotations in the new element are formulated in terms of Updated Lagrangian framework. Consistently coupled polynomial displacement fields are used to satisfy the membrane locking-free condition and the requirement of being able to recover the inextensible bending modes. Quintic transverse displacement interpolation functions are used to represent the bending deformation of the beam, while the axial and torsional displacement fields are derived by integration of the presumably linear membrane and torsional shear strain fields, which are coupled with the transverse displacement fields. Numerical results are presented to demonstrate the superior accuracy and the high convergence rate of the newly developed curved beam element. The stability and accuracy of the new element are further validated by experiments of an instrumented free-swinging steel cable experiencing slack and low tension. Good agreements in cable position and tension are observed between the experimental results and the finite element predictions.

Advances in the formulation of the rotation-free basic shell triangle

A family of rotation-free three node triangular shell elements is presented. The simplest element of the family is based on an assumed constant curvature field expressed in terms of the nodal deflections of a patch of four elements and a constant membrane field computed from the standard linear interpolation of the displacements within each triangle. An enhanced version of the element is obtained by using a quadratic interpolation of the geometry in terms of the six patch nodes. This allows to compute an assumed linear membrane strain field which improves the in-plane behaviour of the original element. A simple and economic version of the element using a single integration point is presented. The efficiency of the different rotation-free shell triangles is demonstrated in many examples of application including linear and non-linear analysis of shells under static and dynamic loads, the inflation and de-inflation of membranes and a sheet stamping problem.

ANALYSIS OF THREE-DIMENSIONAL LOCKING-FREE CURVED BEAM ELEMENT

Most existing curved beam elements suffer from poor convergence difficulties and a heavy computational burden while limit themselves to 2D problems. In this paper, we address and overcome these difficulties by developing a new three-noded locking-free 3D curved beam element. The element formulations, which employ coupled consistent polynomial displacement fields, satisfy the membrane locking-free requirement of being able to recover the inextensible bending mode of the curved beam. Quintic transverse displacement interpolation functions are used to represent the bending deformation of the beam, while the axial and torsional displacement fields are derived by integration of the presumably linear membrane and torsional shear strain fields, which are coupled with the transverse displacement fields. Numerical results of two- and three-dimensional applications are presented to demonstrate the superior accuracy and high convergence rate of the newly developed curved beam element compared with existing ones.

Structural intensity study of plates under low-velocity impact

The transmissions of transient energy flow and dynamic transient response of plate structures under low-velocity impact are presented. The structural intensity approach is used to study the transient dynamic characteristics of plate structures under low-velocity impact. In the dynamic impact response analysis, nine-node degenerated shell elements with assumed shear and membrane strain fields are adopted to model the target and impactor. The dynamic contact-impact algorithm and the governing equations for both the target and impactor are derived based on the updated Lagrangian approach. Explicit integration algorithm has been adopted in the time integration process. The novel structural intensity streamline representation is introduced to interpret energy flow paths for transient dynamic response of plates under low-velocity impact. The effects of plates with and without structural damping on the energy flow and energy path are discussed. Numerical results, including contact force, deflection histories and transient energy flow vectors as well as structural intensity streamlines, show that the present method and representation are an efficient approach for exploring dynamic response for plate structures subjected to low-velocity impact.