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Hyperelasticity
A hyperelastic or Green elastic materialR.W. Ogden, 1984, ''Non-Linear Elastic Deformations'', , Dover. is a type of constitutive model for ideally elastic material for which the stress–strain relationship derives from a strain energy density function. The hyperelastic material is a special case of a Cauchy elastic material. For many materials, linear elastic models do not accurately describe the observed material behaviour. The most common example of this kind of material is rubber, whose stress-strain relationship can be defined as non-linearly elastic, isotropic and incompressible. Hyperelasticity provides a means of modeling the stress–strain behavior of such materials. The behavior of unfilled, vulcanized elastomers often conforms closely to the hyperelastic ideal. Filled elastomers and biological tissues are also often modeled via the hyperelastic idealization. Ronald Rivlin and Melvin Mooney developed the first hyperelastic models, the Neo-Hookean and Mooney– ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elastic (solid Mechanics)
In physics and materials science, elasticity is the ability of a Physical object, body to resist a distorting influence and to return to its original size and shape when that influence or force is removed. Solid objects will Deformation (engineering), deform when adequate Structural load, loads are applied to them; if the material is elastic, the object will return to its initial shape and size after removal. This is in contrast to Plasticity (physics), ''plasticity'', in which the object fails to do so and instead remains in its deformed state. The physical reasons for elastic behavior can be quite different for different materials. In metals, the Crystal structure, atomic lattice changes size and shape when forces are applied (energy is added to the system). When forces are removed, the lattice goes back to the original lower energy state. For rubber elasticity, rubbers and other polymers, elasticity is caused by the stretching of polymer chains when forces are applied. Hoo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mooney–Rivlin Solid
In continuum mechanics, a Mooney–Rivlin solidMooney, M., 1940, ''A theory of large elastic deformation'', Journal of Applied Physics, 11(9), pp. 582–592.Rivlin, R. S., 1948, ''Large elastic deformations of isotropic materials. IV. Further developments of the general theory'', Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 241(835), pp. 379–397. is a hyperelastic material model where the strain energy density function W\, is a linear combination of two invariants of the left Cauchy–Green deformation tensor \boldsymbol. The model was proposed by Melvin Mooney in 1940 and expressed in terms of invariants by Ronald Rivlin in 1948. The strain energy density function for an incompressible Mooney–Rivlin material is :W = C_ (\bar_1-3) + C_ (\bar_2-3), \, where C_ and C_ are empirically determined material constants, and \bar I_1 and \bar I_2 are the first and the second invariant of \bar \boldsymbol B = (\det \boldsy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Drucker Stability
Drucker stability (also called the Drucker stability postulates) refers to a set of mathematical criteria that restrict the possible nonlinear stress-strain relations that can be satisfied by a solid material. The postulates are named after Daniel C. Drucker. A material that does not satisfy these criteria is often found to be unstable in the sense that application of a load to a material point can lead to arbitrary deformations at that material point unless an additional length or time scale is specified in the constitutive relations. The Drucker stability postulates are often invoked in nonlinear finite element analysis. Materials that satisfy these criteria are generally well-suited for numerical analysis, while materials that fail to satisfy this criterion are likely to present difficulties (i.e. non-uniqueness or singularity) during the solution process. Drucker's first stability criterion Drucker's first stability criterion (first proposed by Rodney Hill and also called Hi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Van Der Waals (hyperelatic Model)
Van der Waals or Van der Waal may refer to: People * Fransje van der Waals (born 1950), Dutch medical physician * Grace VanderWaal (born 2004), American singer-songwriter * Henk van der Waal (born 1960), Dutch poet * Joan van der Waals (1920–2022), Dutch physicist * Johannes Diderik van der Waals (1837–1923), Dutch physicist * (1912–1950), Dutch spy, in German service during World War II (see Dutch resistance) Physics There are a series of subjects named after Johannes Diderik van der Waals: * Van der Waals force * Van der Waals equation * Van der Waals molecule * Van der Waals radius * Van der Waals surface Other uses * Van der Waals (crater), named after the physicist * Mona Vanderwaal, ''Pretty Little Liars'' character See also *Van der Wal Van der Wal (or van de Wal, Vander Wal, Vanderwal, van de Wall, VanderWaal) is a toponymic surname of Dutch origin. The original bearer of the name may have lived or worked at or near a "wal": a river embankment, quay, or rampart. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gent (hyperelastic Model)
The Gent hyperelastic material model is a phenomenological model of rubber elasticity that is based on the concept of limiting chain extensibility. In this model, the strain energy density function is designed such that it has a singularity when the first invariant of the left Cauchy-Green deformation tensor reaches a limiting value I_m. The strain energy density function for the Gent model is Gent, A.N., 1996, '' A new constitutive relation for rubber'', Rubber Chemistry Tech., 69, pp. 59-61. : W = -\cfrac \ln\left(1 - \cfrac\right) where \mu is the shear modulus and J_m = I_m -3. In the limit where I_m \rightarrow \infty, the Gent model reduces to the Neo-Hookean solid model. This can be seen by expressing the Gent model in the form : W =- \cfrac\ln\left - (I_1-3)x\right~;~~ x := \cfrac A Taylor series expansion of \ln\left - (I_1-3)x\right/math> around x = 0 and taking the limit as x\rightarrow 0 leads to : W = \cfrac (I_1-3) which is the expression for the st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rubber Elasticity
Rubber elasticity refers to a property of crosslinked rubber: it can be stretched by up to a factor of 10 from its original length and, when released, returns very nearly to its original length. This can be repeated many times with no apparent degradation to the rubber. Rubber is a member of a larger class of materials called elastomers and it is difficult to overestimate their economic and technological importance. Elastomers have played a key role in the development of new technologies in the 20th century and make a substantial contribution to the global economy. Rubber elasticity is produced by several complex molecular processes and its explanation requires a knowledge of advanced mathematics, chemistry and statistical physics, particularly the concept of entropy. Entropy may be thought of as a measure of the thermal energy that is stored in a molecule. Common rubbers, such as polybutadiene and polyisoprene (also called natural rubber), are produced by a process called pol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Marlow (hyperelastic Model)
Marlow may refer to: Places Australia *Marlow, New South Wales, a suburb on the Central Coast Germany *Marlow, Germany United Kingdom *Little Marlow, Buckinghamshire *Marlow, Buckinghamshire **Marlow Bridge, an old suspension bridge over the River Thames **Marlow Rugby Union Football Club, a Rugby Union Club in Buckinghamshire **Marlow F.C., a football club in Buckinghamshire ** Marlow United F.C., a football club in Buckinghamshire **Marlow Regatta, an international rowing event ** Marlow Town Regatta and Festival, a local rowing event and festival *Marlow, Herefordshire United States *Marlow, Missouri *Marlow, New Hampshire *Marlow, Oklahoma *Marlow, Tennessee *Marlow Heights, Maryland People * Marlow (surname), including list of persons and fictional characters with the name Other uses * Marlow Industries See also *Marlowe (other) Marlowe may refer to: Name * Christopher Marlowe (1564–1593), English dramatist, poet and translator * Philip Marlowe, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yeoh (hyperelastic Model)
image:Yeoh model comp.png, 300px, Yeoh model prediction versus experimental data for natural rubber. Model parameters and experimental data froPolymerFEM.com] The Yeoh hyperelastic material modelYeoh, O. H., 1993, "Some forms of the strain energy function for rubber," ''Rubber Chemistry and technology'', Volume 66, Issue 5, November 1993, Pages 754-771. is a phenomenological model for the deformation of nearly incompressible, nonlinear elastic materials such as rubber. The model is based on Ronald Rivlin's observation that the elastic properties of rubber may be described using a strain energy density function which is a power series in the strain invariants I_1, I_2, I_3 of the Cauchy-Green deformation tensors.Rivlin, R. S., 1948, "Some applications of elasticity theory to rubber engineering", in ''Collected Papers of R. S. Rivlin vol. 1 and 2'', Springer, 1997. The Yeoh model for incompressible rubber is a function only of I_1. For compressible rubbers, a dependence on I_3 is ad ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial (hyperelastic Model)
The polynomial hyperelastic material model is a phenomenological model of rubber elasticity. In this model, the strain energy density function is of the form of a polynomial in the two invariants I_1,I_2 of the left Cauchy-Green deformation tensor. The strain energy density function for the polynomial model is Rivlin, R. S. and Saunders, D. W., 1951, '' Large elastic deformations of isotropic materials VII. Experiments on the deformation of rubber.'' Phi. Trans. Royal Soc. London Series A, 243(865), pp. 251-288. : W = \sum_^n C_ (I_1 - 3)^i (I_2 - 3)^j where C_ are material constants and C_=0. For compressible materials, a dependence of volume is added : W = \sum_^n C_ (\bar_1 - 3)^i (\bar_2 - 3)^j + \sum_^m D_(J-1)^ where : \begin \bar_1 & = J^~I_1 ~;~~ I_1 = \lambda_1^2 + \lambda_2 ^2+ \lambda_3 ^2 ~;~~ J = \det(\boldsymbol) \\ \bar_2 & = J^~I_2 ~;~~ I_2 = \lambda_1^2 \lambda_2^2 + \lambda_2^2 \lambda_3^2 + \lambda_3^2 \lambda_1^2 \end In the limit wher ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Soft Tissue
Soft tissue is all the tissue in the body that is not hardened by the processes of ossification or calcification such as bones and teeth. Soft tissue connects, surrounds or supports internal organs and bones, and includes muscle, tendons, ligaments, fat, fibrous tissue, lymph and blood vessels, fasciae, and synovial membranes. with :q=a_E_E_ \qquad Q=b_E_E_ quadratic forms of Green-Lagrange strains E_ and a_, b_ and c material constants. W is the strain energy function per volume unit, which is the mechanical strain energy for a given temperature. Isotropic simplification The Fung-model, simplified with isotropic hypothesis (same mechanical properties in all directions). This written in respect of the principal stretches (\lambda_i): :W = \frac\left (\lambda_1^2 + \lambda_2^2 + \lambda_3^2 - 3) + b\left( e^ -1 \right) \right/math> , where a, b and c are constants. Simplification for small and big stretches For small strains, the exponential term is very small, thus neg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Phenomenological Model
A phenomenological model is a scientific model that describes the empirical relationship of phenomena to each other, in a way which is consistent with fundamental theory, but is not directly derived from theory. In other words, a phenomenological model is not derived from first principles. A phenomenological model forgoes any attempt to explain why the variables interact the way they do, and simply attempts to describe the relationship, with the assumption that the relationship extends past the measured values. Regression analysis is sometimes used to create statistical models that serve as phenomenological models. Examples of use Phenomenological models have been characterized as being completely independent of theories, though many phenomenological models, while failing to be derivable from a theory, incorporate principles and laws associated with theories. The liquid drop model of the atomic nucleus, for instance, portrays the nucleus as a liquid drop and describes it as having ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
