The Compatibility Constraint in Linear Elasticity

2000 ◽  
pp. 393-398
Author(s):  
P. Podio-Guidugli
Keyword(s):  
Linear Elasticity ◽  
Compatibility Constraint

Computing cell tractions from particle displacements on silicone rubber substrata

1994 ◽  
Vol 52 ◽  
pp. 162-163
Author(s):  
Tim Oliver ◽  
Akira Ishihara ◽  
Ken Jacobsen ◽  
Micah Dembo
Keyword(s):  
Plane Stress ◽  
Linear Elasticity ◽  
Silicone Rubber ◽  
Mathematical Analysis ◽  
Elastic Recovery ◽  
Video Images ◽  
Cell Traction Forces ◽  
Latex Beads ◽  
Cell Traction ◽  
Better Than

In order to better understand the distribution of cell traction forces generated by rapidly locomoting cells, we have applied a mathematical analysis to our modified silicone rubber traction assay, based on the plane stress Green’s function of linear elasticity. To achieve this, we made crosslinked silicone rubber films into which we incorporated many more latex beads than previously possible (Figs. 1 and 6), using a modified airbrush. These films could be deformed by fish keratocytes, were virtually drift-free, and showed better than a 90% elastic recovery to micromanipulation (data not shown). Video images of cells locomoting on these films were recorded. From a pair of images representing the undisturbed and stressed states of the film, we recorded the cell’s outline and the associated displacements of bead centroids using Image-1 (Fig. 1). Next, using our own software, a mesh of quadrilaterals was plotted (Fig. 2) to represent the cell outline and to superimpose on the outline a traction density distribution. The net displacement of each bead in the film was calculated from centroid data and displayed with the mesh outline (Fig. 3).

Mathematical analysis and simulation of fixed point formulation of Cauchy problem in linear elasticity

2021 ◽  
Vol 187 ◽  
pp. 231-247
Author(s):  
Abdellatif Ellabib ◽  
Abdeljalil Nachaoui ◽  
Abdessamad Ousaadane
Keyword(s):  
Cauchy Problem ◽  
Fixed Point ◽  
Linear Elasticity ◽  
Mathematical Analysis

Locking-Free and Gradient-Robust $${\varvec{H}}({{\,\mathrm{{\text {div}}}\,}})$$-Conforming HDG Methods for Linear Elasticity

2021 ◽  
Vol 86 (3) ◽  
Author(s):  
Guosheng Fu ◽  
Christoph Lehrenfeld ◽  
Alexander Linke ◽  
Timo Streckenbach
Keyword(s):  
Linear Elasticity

scholarly journals A linear-elasticity-based mesh moving method with no cycle-to-cycle accumulated distortion

2021 ◽  
Author(s):  
Patrícia Tonon ◽  
Rodolfo André Kuche Sanches ◽  
Kenji Takizawa ◽  
Tayfun E. Tezduyar
Keyword(s):  
Linear Elasticity ◽  
Solid Surfaces ◽  
Moving Mesh ◽  
Test Case ◽  
Half Cycle ◽  
Arbitrary Lagrangian Eulerian ◽  
Interface Motion ◽  
Mesh Motion ◽  
Ale Methods ◽  
Moving Mesh Methods

AbstractGood mesh moving methods are always part of what makes moving-mesh methods good in computation of flow problems with moving boundaries and interfaces, including fluid–structure interaction. Moving-mesh methods, such as the space–time (ST) and arbitrary Lagrangian–Eulerian (ALE) methods, enable mesh-resolution control near solid surfaces and thus high-resolution representation of the boundary layers. Mesh moving based on linear elasticity and mesh-Jacobian-based stiffening (MJBS) has been in use with the ST and ALE methods since 1992. In the MJBS, the objective is to stiffen the smaller elements, which are typically placed near solid surfaces, more than the larger ones, and this is accomplished by altering the way we account for the Jacobian of the transformation from the element domain to the physical domain. In computing the mesh motion between time levels $$t_n$$ t n and $$t_{n+1}$$ t n + 1 with the linear-elasticity equations, the most common option is to compute the displacement from the configuration at $$t_n$$ t n . While this option works well for most problems, because the method is path-dependent, it involves cycle-to-cycle accumulated mesh distortion. The back-cycle-based mesh moving (BCBMM) method, introduced recently with two versions, can remedy that. In the BCBMM, there is no cycle-to-cycle accumulated distortion. In this article, for the first time, we present mesh moving test computations with the BCBMM. We also introduce a version we call “half-cycle-based mesh moving” (HCBMM) method, and that is for computations where the boundary or interface motion in the second half of the cycle consists of just reversing the steps in the first half and we want the mesh to behave the same way. We present detailed 2D and 3D test computations with finite element meshes, using as the test case the mesh motion associated with wing pitching. The computations show that all versions of the BCBMM perform well, with no cycle-to-cycle accumulated distortion, and with the HCBMM, as the wing in the second half of the cycle just reverses its motion steps in the first half, the mesh behaves the same way.

Hydrogen-enhanced compatibility constraint for intergranular failure in FCC FeNiCoCrMn high-entropy alloy

2021 ◽  
Vol 184 ◽  
pp. 109407
Author(s):  
K.M. Bertsch ◽  
K.E. Nygren ◽  
S. Wang ◽  
H. Bei ◽  
A. Nagao
Keyword(s):  
High Entropy Alloy ◽  
High Entropy ◽  
Intergranular Failure ◽  
Compatibility Constraint

scholarly journals Mathematical models of supersonic and intersonic crack propagation in linear elastodynamics

2021 ◽  
Author(s):  
Javier Bonet ◽  
Antonio J. Gil
Keyword(s):  
Kinetic Energy ◽  
Crack Propagation ◽  
Mathematical Models ◽  
Linear Elasticity ◽  
Strain Energy ◽  
Equations Of Motion ◽  
Two Dimensions ◽  
Discontinuous Solutions ◽  
Linear Elasticity Theory ◽  
Intersonic Crack Propagation

AbstractThis paper presents mathematical models of supersonic and intersonic crack propagation exhibiting Mach type of shock wave patterns that closely resemble the growing body of experimental and computational evidence reported in recent years. The models are developed in the form of weak discontinuous solutions of the equations of motion for isotropic linear elasticity in two dimensions. Instead of the classical second order elastodynamics equations in terms of the displacement field, equivalent first order equations in terms of the evolution of velocity and displacement gradient fields are used together with their associated jump conditions across solution discontinuities. The paper postulates supersonic and intersonic steady-state crack propagation solutions consisting of regions of constant deformation and velocity separated by pressure and shear shock waves converging at the crack tip and obtains the necessary requirements for their existence. It shows that such mathematical solutions exist for significant ranges of material properties both in plane stress and plane strain. Both mode I and mode II fracture configurations are considered. In line with the linear elasticity theory used, the solutions obtained satisfy exact energy conservation, which implies that strain energy in the unfractured material is converted in its entirety into kinetic energy as the crack propagates. This neglects dissipation phenomena both in the material and in the creation of the new crack surface. This leads to the conclusion that fast crack propagation beyond the classical limit of the Rayleigh wave speed is a phenomenon dominated by the transfer of strain energy into kinetic energy rather than by the transfer into surface energy, which is the basis of Griffiths theory.

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