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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] |
Solid Mechanics
Solid mechanics, also known as mechanics of solids, is the branch of continuum mechanics that studies the behavior of solid materials, especially their motion and deformation under the action of forces, temperature changes, phase changes, and other external or internal agents. Solid mechanics is fundamental for civil, aerospace, nuclear, biomedical and mechanical engineering, for geology, and for many branches of physics such as materials science. It has specific applications in many other areas, such as understanding the anatomy of living beings, and the design of dental prostheses and surgical implants. One of the most common practical applications of solid mechanics is the Euler–Bernoulli beam equation. Solid mechanics extensively uses tensors to describe stresses, strains, and the relationship between them. Solid mechanics is a vast subject because of the wide range of solid materials available, such as steel, wood, concrete, biological materials, textiles, geological ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperelastic Material
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–Ri ... [...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] |
Strain Energy Density Function
A strain energy density function or stored energy density function is a scalar-valued function that relates the strain energy density of a material to the deformation gradient. : W = \hat(\boldsymbol) = \hat(\boldsymbol^T\cdot\boldsymbol) =\bar(\boldsymbol) = \bar(\boldsymbol^\cdot\boldsymbol)=\tilde(\boldsymbol,\boldsymbol) Equivalently, : W = \hat(\boldsymbol) = \hat(\boldsymbol^T\cdot\boldsymbol\cdot\boldsymbol) =\tilde(\boldsymbol,\boldsymbol) where \boldsymbol is the (two-point) deformation gradient tensor, \boldsymbol is the right Cauchy–Green deformation tensor, \boldsymbol is the left Cauchy–Green deformation tensor, and \boldsymbol is the rotation tensor from the polar decomposition of \boldsymbol. For an anisotropic material, the strain energy density function \hat(\boldsymbol) depends implicitly on reference vectors or tensors (such as the initial orientation of fibers in a composite) that characterize internal material texture. The spatial representat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neo-Hookean Solid
A neo-Hookean solid is a hyperelastic material model, similar to Hooke's law, that can be used for predicting the nonlinear stress-strain behavior of materials undergoing large deformations. The model was proposed by Ronald Rivlin in 1948. In contrast to linear elastic materials, the stress-strain curve of a neo-Hookean material is not linear. Instead, the relationship between applied stress and strain is initially linear, but at a certain point the stress-strain curve will plateau. The neo-Hookean model does not account for the dissipative release of energy as heat while straining the material and perfect elasticity is assumed at all stages of deformation. The neo-Hookean model is based on the statistical thermodynamics of cross-linked polymer chains and is usable for plastics and rubber-like substances. Cross-linked polymers will act in a neo-Hookean manner because initially the polymer chains can move relative to each other when a stress is applied. However, at a certain ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compressible
In thermodynamics and fluid mechanics, the compressibility (also known as the coefficient of compressibility or, if the temperature is held constant, the isothermal compressibility) is a measure of the instantaneous relative volume change of a fluid or solid as a response to a pressure (or mean stress) change. In its simple form, the compressibility \kappa (denoted in some fields) may be expressed as :\beta =-\frac\frac, where is volume and is pressure. The choice to define compressibility as the negative of the fraction makes compressibility positive in the (usual) case that an increase in pressure induces a reduction in volume. The reciprocal of compressibility at fixed temperature is called the isothermal bulk modulus. Definition The specification above is incomplete, because for any object or system the magnitude of the compressibility depends strongly on whether the process is isentropic or isothermal. Accordingly, isothermal compressibility is defined: :\beta_T=-\frac ... [...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] |
Hyperelastic Material
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–Ri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strain Energy Density Function
A strain energy density function or stored energy density function is a scalar-valued function that relates the strain energy density of a material to the deformation gradient. : W = \hat(\boldsymbol) = \hat(\boldsymbol^T\cdot\boldsymbol) =\bar(\boldsymbol) = \bar(\boldsymbol^\cdot\boldsymbol)=\tilde(\boldsymbol,\boldsymbol) Equivalently, : W = \hat(\boldsymbol) = \hat(\boldsymbol^T\cdot\boldsymbol\cdot\boldsymbol) =\tilde(\boldsymbol,\boldsymbol) where \boldsymbol is the (two-point) deformation gradient tensor, \boldsymbol is the right Cauchy–Green deformation tensor, \boldsymbol is the left Cauchy–Green deformation tensor, and \boldsymbol is the rotation tensor from the polar decomposition of \boldsymbol. For an anisotropic material, the strain energy density function \hat(\boldsymbol) depends implicitly on reference vectors or tensors (such as the initial orientation of fibers in a composite) that characterize internal material texture. The spatial representat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Strain Theory
In continuum mechanics, the finite strain theory—also called large strain theory, or large deformation theory—deals with deformations in which strains and/or rotations are large enough to invalidate assumptions inherent in infinitesimal strain theory. In this case, the undeformed and deformed configurations of the continuum are significantly different, requiring a clear distinction between them. This is commonly the case with elastomers, plastically-deforming materials and other fluids and biological soft tissue. Displacement The displacement of a body has two components: a rigid-body displacement and a deformation. * A rigid-body displacement consists of a simultaneous translation (physics) and rotation of the body without changing its shape or size. * Deformation implies the change in shape and/or size of the body from an initial or undeformed configuration \kappa_0(\mathcal B) to a current or deformed configuration \kappa_t(\mathcal B) (Figure 1). A change in the conf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stress Measures
In continuum mechanics, the most commonly used measure of stress is the Cauchy stress tensor, often called simply ''the'' stress tensor or "true stress". However, several alternative measures of stress can be defined: #The Kirchhoff stress (\boldsymbol). #The Nominal stress (\boldsymbol). #The first Piola–Kirchhoff stress (\boldsymbol). This stress tensor is the transpose of the nominal stress (\boldsymbol = \boldsymbol^T). #The second Piola–Kirchhoff stress or PK2 stress (\boldsymbol). #The Biot stress (\boldsymbol) Definitions Consider the situation shown in the following figure. The following definitions use the notations shown in the figure. In the reference configuration \Omega_0, the outward normal to a surface element d\Gamma_0 is \mathbf \equiv \mathbf_0 and the traction acting on that surface (assuming it deforms like a generic vector belonging to the deformation) is \mathbf_0 leading to a force vector d\mathbf_0. In the deformed configuration \Omega, the surfac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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