An elastic singularity in joined half-spaces

Walpole, L. J.

doi:10.1016/0020-7225(95)00120-4

Cited by 23 publications

(18 citation statements)

References 20 publications

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“…In this way, cells can actively sense not only the presence of a close-by surface, but also its shape and boundary conditions. To predict the effect of boundaries on cell organization, we study a semiinfinite space with a planar surface, for which the elastic equations can be solved exactly (23). The details of the boundary conditions in a physiological context can be very complicated.…”

Section: Resultsmentioning

confidence: 99%

Cell organization in soft media due to active mechanosensing

Bischofs

Schwarz

2003

Proc. Natl. Acad. Sci. U.S.A.

288

263

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Adhering cells actively probe the mechanical properties of their environment and use the resulting information to position and orient themselves. We show that a large body of experimental observations can be consistently explained from one unifying principle, namely that cells strengthen contacts and cytoskeleton in the direction of large effective stiffness. Using linear elasticity theory to model the extracellular environment, we calculate optimal cell organization for several situations of interest and find excellent agreement with experiments for fibroblasts, both on elastic substrates and in collagen gels: cells orient in the direction of external tensile strain; they orient parallel and normal to free and clamped surfaces, respectively; and they interact elastically to form strings. Our method can be applied for rational design of tissue equivalents. Moreover, our results indicate that the concept of contact guidance has to be reevaluated. We also suggest that cell-matrix contacts are up-regulated by large effective stiffness in the environment because, in this way, build-up of force is more efficient.T he mechanical activity of adherent cells usually is attributed to their physiological function. For example, fibroblasts are believed to maintain the structural integrity of connective tissue and to participate in wound healing by actively pulling on their environment. During recent years, it has become clear that there is another important role for mechanical activity of adherent cells: by pulling on their environment, cells can actively sense its mechanical properties and react to it in a specific way (1-3). Harris et al. (4) observed surprisingly large tension fields for fibroblasts on elastic substrates, which induce mechanical activity of other cells, even when located at considerable distance. When plated on elastic substrates of increased rigidity, many cell types show increased spreading and better developed stress fibers and focal adhesions (5). Fibroblasts on elastic substrates orient in the direction of tensile strain (6) and locomote in favor of regions of larger rigidity or tensile strain (7). The same response has been reported for vascular smooth muscle cells on rigidity gradients (8). Similar observations have been reported numerous times also for tissue cells in hydrogels. For fibroblasts in collagen gels, Bell et al. (9) not only found that traction considerably contracts the gel, but also reported orientational effects: cells align along the direction of pull between fixed points and parallel to free surfaces. When a collagen gel is stretched uniaxially, cells orient in the direction of principal strain (10). Moreover, cells align in a nose-to-tail configuration, thus forming strings running in parallel to the direction of external strain. If a collagen gel is cut perpendicular to the direction of tensile strain and if cells are present in sufficient numbers, they round up and reorient parallel to the free surface introduced (11).The response of adherent animal cells to mechanical input ...

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Section: Resultsmentioning

confidence: 99%

Cell organization in soft media due to active mechanosensing

Bischofs

Schwarz

2003

Proc. Natl. Acad. Sci. U.S.A.

288

263

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“…where the coefficients M, J, C, B depend on the boundary condition (subscripts: free f , clamped c) and the Poisson ratio ν [39]:…”

Section: Dipoles In Elastic Halfspacementioning

confidence: 99%

Elastic interactions of active cells with soft materials

2004

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Anchorage-dependent cells collect information on the mechanical properties of the environment through their contractile machineries and use this information to position and orient themselves. Since the probing process is anisotropic, cellular force patterns during active mechanosensing can be modelled as anisotropic force contraction dipoles. Their build-up depends on the mechanical properties of the environment, including elastic rigidity and prestrain. In a finite sized sample, it also depends on sample geometry and boundary conditions through image strain fields.We discuss the interactions of active cells with an elastic environment and compare it to the case of physical force dipoles. Despite marked differences, both cases can be described in the same theoretical framework. We exactly solve the elastic equations for anisotropic force contraction dipoles in different geometries (full space, halfspace and sphere) and with different boundary conditions. These results are then used to predict optimal position and orientation of mechanosensing cells in soft material.

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“…When a uniform normal or shear stress is applied on the surface of a half-space containing particles, the local stress field can be calculated by applying the fundamental solutions for Mindlin's problem [33][34][35][36][37] …”

Section: Introductionmentioning

confidence: 99%

“…In what follows, §2 reviews fundamental solutions for a concentrated force in a semiinfinite domain [37] and introduces an explicit form Green's function for Mindlin's problem [33]. Using the Green function technique [25], we derive the elastic field for a semi-infinite solid containing inclusions.…”

Section: Introductionmentioning

confidence: 99%

“…However, when an inhomogeneity is subjected to a body force or a far-field stress, the mismatch between inhomogeneity and the matrix can be simulated by an appropriately chosen eigenstrain on the subdomain so that the elastic field can be obtained through solving the inclusion problem [25,40]. Notice that the inclusion problems for a semi-infinite domain or two half-infinite domains have been studied [36,37,41,42]. However, the inhomogeneity problem for a semi-infinite domain has not been solved in the literature yet.…”

Section: Introductionmentioning

confidence: 99%

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Boundary effect on the elastic field of a semi-infinite solid containing inhomogeneities

2015

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The boundary effect of one inhomogeneity embedded in a semi-infinite solid at different depths has firstly been investigated using the fundamental solution for Mindlin's problem. Expanding the eigenstrain in a polynomial form and using the Eshelby's equivalent inclusion method, one can calculate the eigenstrain and thus obtain the elastic field. When the inhomogeneity is far from the boundary, the solution recovers Eshelby's solution. The method has been extended to a many-particle system in a semi-infinite solid, which is first demonstrated by the cases of two spheres. The comparison of the asymptotic form solution with the finite-element results shows the accuracy and capability of this method. The solution has been used to illustrate the boundary effects on its effective material behaviour of a semi-infinite simple cubic lattice particulate composite. The local field of a semi-infinite composite has been calculated at different volume fractions. A representative unit cell has been taken with different depths to the surface. The average stress and strain of the unit cell have been calculated under uniform loading conditions of normal or shear force on the surface, respectively. The effective elastic moduli of the unit cell not only depend on the material proportion, but also on its distance to the surface. The present model can be extended to other types of particle distribution and ellipsoidal particles.

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An elastic singularity in joined half-spaces

Cited by 23 publications

References 20 publications

Cell organization in soft media due to active mechanosensing

Cell organization in soft media due to active mechanosensing

Elastic interactions of active cells with soft materials

Boundary effect on the elastic field of a semi-infinite solid containing inhomogeneities

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