Compressed elastic films on soft substrates release part of their strain energy by wrinkling, which represents a loss of symmetry, characterized by a pitchfork bifurcation. Its development is well ...understood at the onset of supercritical bifurcation, but not beyond, or in the case of subcritical bifurcation. This is mainly due to nonlinearities and the extreme imperfection sensitivity. In both types of bifurcations, the energy–displacement diagrams that can characterize an energy landscape are non-convex, which is notoriously difficult to determine numerically or experimentally, let alone analytically. To gain an elementary understanding of such potential energy landscapes, we take a thin beam theory suitable for analyzing large displacements under small strains and significantly reduce its complexity by reformulating it in terms of the tangent rotation angle. This enables a comprehensive analytical and numerical analysis of wrinkling elastic films on planar substrates, which are effective stiffening and/or softening due to either geometric or material nonlinearities. We also validate our findings experimentally. We explicitly show how effective stiffening nonlinear behavior (e.g., due to substrate or membrane deformations) leads to a supercritical post-bifurcation response, makes the energy landscape non-convex through energy barriers causing multistability, which is extremely problematic for numerical computation. Moreover, this type of nonlinearity promotes uni-modal, uniformly distributed, periodic deformation patterns. In contrast, nonlinear effective softening behavior leads to subcritical post-bifurcation behavior, similarly divides the energy landscape by energy barriers and conversely promotes localization of deformations. With our theoretical model we can thus explain an experimentally observed phenomenon that in structures with effective softening followed by an effective stiffening behavior, the symmetry is initially broken by localizing the deformation and later restored by forming periodic, distributed deformation patterns as the load is increased. Finally, we show that initial imperfections can significantly alter the local or global energy-minimizing deformation pattern and completely remove some energy barriers. We envision that this knowledge can be extrapolated and exploited to convexify extremely divergent energy landscapes of more sophisticated systems, such as wrinkling compressed films on curved substrates (e.g., on cylinders and spheres) and that it will enable elementary analysis and the development of specialized numerical tools.
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•Effective softening/stiffening causes localized/distributed deformation.•Imperfections lower some energy minima and remove energy barriers.•Effective softening followed by stiffening causes loss and recovery of periodic symmetry.•Removal of effective stiffening/softening terms convexifies the energy landscape.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
We construct planar bi-Sobolev mappings whose local volume distortion is bounded from below by a given function f∈Lp with p>1. More precisely, for any 1<q<(p+1)/2 we construct W1,q-bi-Sobolev maps ...with identity boundary conditions; for f∈L∞, we provide bi-Lipschitz maps. The basic building block of our construction are bi-Lipschitz maps which stretch a given compact subset of the unit square by a given factor while preserving the boundary. The construction of these stretching maps relies on a slight strengthening of the celebrated covering result of Alberti, Csörnyei, and Preiss for measurable planar sets in the case of compact sets. We apply our result to a model functional in nonlinear elasticity, the integrand of which features fast blowup as the Jacobian determinant of the deformation becomes small. For such functionals, the derivation of the equilibrium equations for minimizers requires an additional regularization of test functions, which our maps provide.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
Soft structures are capable of undergoing reversible large strains and deformations when facing different types of loadings. Due to the limitations of linear elastic models, researchers have ...developed and employed different nonlinear elastic models capable of accurately modelling large deformations and strains. These models are significantly different in formulation and application. As hyperelastic strain energy density models provide researchers with a good fit for the mechanical behaviour of biological tissues, research studies on using these constitutive models together with different continuum-mechanics-based formulations have reached notable outcomes. With the improvements in biomechanical devices, in-vivo and in-vitro studies have increased significantly in the past few years which emphasises the importance of reviewing the latest works in this field. Besides, since soft structures are used for different mechanical and biomechanical applications such as prosthetics, soft robots, packaging, and wearing devices, the application of a proper hyperelastic strain energy density law in modelling the structure is of high importance. Therefore, in this review, a detailed classified analysis of the mechanics of hyperelastic structures is presented by focusing on the application of different hyperelastic strain energy density models. Previous studies on biological soft parts of the body (brain, artery, cartilage, liver, skeletal muscle, ligament, skin, tongue, heel pad and adipose tissue) are presented in detail and the hyperelastic strain energy models used for each biological tissue is discussed. Besides, the mechanics (deformation, buckling, inflation, etc.) of polymeric structures in different mechanical conditions is presented using previous studies in this field and the strength of hyperelastic strain energy density models in analysing their mechanics is presented.
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In native states, animal cells of many types are supported by a fibrous network that forms the main structural component of the ECM. Mechanical interactions between cells and the 3D ECM critically ...regulate cell function, including growth and migration. However, the physical mechanism that governs the cell interaction with fibrous 3D ECM is still not known. In this article, we present single-cell traction force measurements using breast tumor cells embedded within 3D collagen matrices. We recreate the breast tumor mechanical environment by controlling the microstructure and density of type I collagen matrices. Our results reveal a positive mechanical feedback loop: cells pulling on collagen locally align and stiffen the matrix, and stiffer matrices, in return, promote greater cell force generation and a stiffer cell body. Furthermore, cell force transmission distance increases with the degree of strain-induced fiber alignment and stiffening of the collagen matrices. These findings highlight the importance of the nonlinear elasticity of fibrous matrices in regulating cell–ECM interactions within a 3D context, and the cell force regulation principle that we uncover may contribute to the rapid mechanical tissue stiffening occurring in many diseases, including cancer and fibrosis.
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Stress controls the mechanics of collagen networks Licup, Albert James; Münster, Stefan; Sharma, Abhinav ...
Proceedings of the National Academy of Sciences - PNAS,
08/2015, Volume:
112, Issue:
31
Journal Article
Peer reviewed
Open access
Collagen is the main structural and load-bearing element of various connective tissues, where it forms the extracellular matrix that supports cells. It has long been known that collagenous tissues ...exhibit a highly nonlinear stress–strain relationship, although the origins of this nonlinearity remain unknown. Here, we show that the nonlinear stiffening of reconstituted type I collagen networks is controlled by the applied stress and that the network stiffness becomes surprisingly insensitive to network concentration. We demonstrate how a simple model for networks of elastic fibers can quantitatively account for the mechanics of reconstituted collagen networks. Our model points to the important role of normal stresses in determining the nonlinear shear elastic response, which can explain the approximate exponential relationship between stress and strain reported for collagenous tissues. This further suggests principles for the design of synthetic fiber networks with collagen-like properties, as well as a mechanism for the control of the mechanics of such networks.
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Based on the classical theory of simple materials of differential type and the results on the analytical form of constitutive models consistent with the laws of thermodynamics, we introduce a very ...general response function for the Cauchy stress tensor of a dispersive hyperelastic solid. Next, by focusing on the propagation of localised waves in slightly dispersive quasi incompressible solids, we prove the existence of a rich variety of solitary wave solutions as well as kink wave solutions. Our analysis and results can be easily specialised to shape memory alloys.
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•We develop a mathematical model for the stress tensor of dispersive hyperelastic solids within the theory of simple materials of differential type.•We study the wave propagation in dispersive hyperelastic solids.•In the small but finite regime, we find exact solutions representing solitary and kink waves.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
10.
A theory of finite deformation magneto-viscoelasticity Saxena, Prashant; Hossain, Mokarram; Steinmann, Paul
International journal of solids and structures,
November 2013, 2013-11-00, 20131101, Volume:
50, Issue:
24
Journal Article
Peer reviewed
Open access
This paper deals with the mathematical modelling of large strain magneto-viscoelastic deformations. Energy dissipation is assumed to occur both due to the mechanical viscoelastic effects as well as ...the resistance offered by the material to magnetisation. Existence of internal damping mechanisms in the body is considered by decomposing the deformation gradient and the magnetic induction into ‘elastic’ and ‘viscous’ parts. Constitutive laws for material behaviour and evolution equations for the non-equilibrium fields are derived that agree with the laws of thermodynamics. To illustrate the theory the problems of stress relaxation, magnetic field relaxation, time dependent magnetic induction and strain are formulated and solved for a specific form of the constitutive law. The results, that show the effect of several modelling parameters on the deformation and magnetisation process, are illustrated graphically.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPCLJ, UPUK, ZAGLJ, ZRSKP
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