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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] |
Scalar (mathematics)
A scalar is an element of a field which is used to define a ''vector space''. In linear algebra, real numbers or generally elements of a field are called scalars and relate to vectors in an associated vector space through the operation of scalar multiplication (defined in the vector space), in which a vector can be multiplied by a scalar in the defined way to produce another vector. Generally speaking, a vector space may be defined by using any field instead of real numbers (such as complex numbers). Then scalars of that vector space will be elements of the associated field (such as complex numbers). A scalar product operation – not to be confused with scalar multiplication – may be defined on a vector space, allowing two vectors to be multiplied in the defined way to produce a scalar. A vector space equipped with a scalar product is called an inner product space. A quantity described by multiple scalars, such as having both direction and magnitude, is called a '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constitutive Equations
In physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities (especially kinetic quantities as related to kinematic quantities) that is specific to a material or substance, and approximates the response of that material to external stimuli, usually as applied fields or forces. They are combined with other equations governing physical laws to solve physical problems; for example in fluid mechanics the flow of a fluid in a pipe, in solid state physics the response of a crystal to an electric field, or in structural analysis, the connection between applied stresses or loads to strains or deformations. Some constitutive equations are simply phenomenological; others are derived from first principles. A common approximate constitutive equation frequently is expressed as a simple proportionality using a parameter taken to be a property of the material, such as electrical conductivity or a spring constant. However, it is o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rubber Properties
Rubber, also called India rubber, latex, Amazonian rubber, ''caucho'', or ''caoutchouc'', as initially produced, consists of polymers of the organic compound isoprene, with minor impurities of other organic compounds. Thailand, Malaysia, and Indonesia are three of the leading rubber producers. Types of polyisoprene that are used as natural rubbers are classified as elastomers. Currently, rubber is harvested mainly in the form of the latex from the rubber tree (''Hevea brasiliensis'') or others. The latex is a sticky, milky and white colloid drawn off by making incisions in the bark and collecting the fluid in vessels in a process called "tapping". The latex then is refined into the rubber that is ready for commercial processing. In major areas, latex is allowed to coagulate in the collection cup. The coagulated lumps are collected and processed into dry forms for sale. Natural rubber is used extensively in many applications and products, either alone or in combination with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuum Mechanics
Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such models in the 19th century. Explanation A continuum model assumes that the substance of the object fills the space it occupies. Modeling objects in this way ignores the fact that matter is made of atoms, and so is not continuous; however, on length scales much greater than that of inter-atomic distances, such models are highly accurate. These models can be used to derive differential equations that describe the behavior of such objects using physical laws, such as mass conservation, momentum conservation, and energy conservation, and some information about the material is provided by constitutive relationships. Continuum mechanics deals with the physical properties of solids and fluids which are independent of any particular coordinate sy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ogden–Roxburgh Model
The Ogden–Roxburgh model is an approach which extends hyperelastic material models to allow for the Mullins effect. It is used in several commercial finite element The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat t ... codes, and is named for R.W. Ogden and D. G. Roxburgh. The basis of pseudo-elastic material models is a hyperelastic second Piola–Kirchhoff stress \boldsymbol_0, which is derived from a suitable strain energy density function W(\boldsymbol): : \boldsymbol = 2 \frac \quad . The key idea of pseudo-elastic material models is that the stress during the first loading process is equal to the basic stress \boldsymbol_0. Upon unloading and reloading \boldsymbol_0 is multiplied by a positive softening function \eta. The function \eta thereby depends on the strain energ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuum Mechanics/Thermoelasticity
Continuum may refer to: * Continuum (measurement), theories or models that explain gradual transitions from one condition to another without abrupt changes Mathematics * Continuum (set theory), the real line or the corresponding cardinal number * Linear continuum, any ordered set that shares certain properties of the real line * Continuum (topology), a nonempty compact connected metric space (sometimes Hausdorff space) * Continuum hypothesis, the hypothesis that no infinite sets are larger than the integers but smaller than the real numbers * Cardinality of the continuum, a cardinal number that represents the size of the set of real numbers Science * Continuum morphology, in plant morphology, underlining the continuum between morphological categories * Continuum concept, in psychology * Continuum mechanics, in physics, deals with continuous matter * Space-time continuum, any mathematical model that combines space and time into a single continuum * Continuum theory of specific ... [...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] |
Arruda–Boyce Model
In continuum mechanics, an Arruda–Boyce model Arruda, E. M. and Boyce, M. C., 1993, A three-dimensional model for the large stretch behavior of rubber elastic materials,, J. Mech. Phys. Solids, 41(2), pp. 389–412. is a hyperelastic constitutive model used to describe the mechanical behavior of rubber and other polymeric substances. This model is based on the statistical mechanics of a material with a cubic representative volume element containing eight chains along the diagonal directions. The material is assumed to be incompressible. The model is named after Ellen Arruda and Mary Cunningham Boyce, who published it in 1993. The strain energy density function for the incompressible Arruda–Boyce model is given byBergstrom, J. S. and Boyce, M. C., 2001, Deformation of Elastomeric Networks: Relation between Molecular Level Deformation and Classical Statistical Mechanics Models of Rubber Elasticity, Macromolecules, 34 (3), pp 614–626, . : W = Nk_B\theta\sqrt\left b ... [...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] |
Ogden (hyperelastic Model)
The Ogden material model is a hyperelastic material model used to describe the non-linear stress–strain behaviour of complex materials such as rubbers, polymers, and biological tissue. The model was developed by Raymond Ogden in 1972.Ogden, R. W., (1972). ''Large Deformation Isotropic Elasticity – On the Correlation of Theory and Experiment for Incompressible Rubberlike Solids'', Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 326, No. 1567 (1 February 1972), pp. 565–584. The Ogden model, like other hyperelastic material models, assumes that the material behaviour can be described by means of a strain energy density function, from which the stress–strain relationships can be derived. Ogden material model In the Ogden material model, the strain energy density is expressed in terms of the principal stretches \,\!\lambda_j, \,\!j=1,2,3 as: : W\left( \lambda_1,\lambda_2,\lambda_3 \right) = \sum_^N \frac\left( \lambda_1^ + \ ... [...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] |
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] |
