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Strain Energy Density Function
A strain energy density function or stored energy density function is a scalar (mathematics), scalar-valued function (mathematics), 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 Finite strain theory#The_right_Cauchy–Green_deformation_tensor, right Cauchy–Green deformation tensor, \boldsymbol is the Finite strain theory#The_left_Cauchy–Green_or_Finger_deformation_tensor, 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(\bolds ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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] |
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Internal Energy
The internal energy of a thermodynamic system is the energy of the system as a state function, measured as the quantity of energy necessary to bring the system from its standard internal state to its present internal state of interest, accounting for the gains and losses of energy due to changes in its internal state, including such quantities as magnetization. It excludes the kinetic energy of motion of the system as a whole and the potential energy of position of the system as a whole, with respect to its surroundings and external force fields. It includes the thermal energy, ''i.e.'', the constituent particles' kinetic energies of motion relative to the motion of the system as a whole. Without a thermodynamic process, the internal energy of an isolated system cannot change, as expressed in the law of conservation of energy, a foundation of the first law of thermodynamics. The notion has been introduced to describe the systems characterized by temperature variations, te ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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. 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 Hevea brasiliensis, Pará 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". Manufacturers refine this latex into the rubber that is ready for commercial processing. Natural rubber is used extensively in many applications and products, either alone or in combination with other materials. In most of its useful forms, it has a large stretch ratio and high resilience and also is buoyant and water-proof. Industrial demand for rubber-like materials began to out ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Continuum Mechanics
Continuum mechanics is a branch of mechanics that deals with the deformation of and transmission of forces through materials modeled as a ''continuous medium'' (also called a ''continuum'') rather than as discrete particles. Continuum mechanics deals with ''deformable bodies'', as opposed to rigid bodies. A continuum model assumes that the substance of the object completely fills the space it occupies. While ignoring the fact that matter is made of atoms, this provides a sufficiently accurate description of matter on length scales much greater than that of inter-atomic distances. The concept of a continuous medium allows for intuitive analysis of bulk matter by using differential equations that describe the behavior of such matter according to physical laws, such as mass conservation, momentum conservation, and energy conservation. Information about the specific material is expressed in constitutive relationships. Continuum mechanics treats the physical properties of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Ogden–Roxburgh Model
The Ogden–Roxburgh model is an approach published in 1999 which extends hyperelastic material models to allow for the Mullins effect. It is used in several commercial finite element codes, and is named after R.W. Ogden and D. G. Roxburgh. The fundamental idea of the approach can already be found in a paper by De Souza Neto ''et al.'' from 1994. 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 A strain energy density function or stored energy density function is a scalar (mathematics), scalar-valued function (mathematics), function that relates the strain energy density of a material to the deformation gradient. : W = \hat(\boldsy ... 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 re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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 (physics), continuous media * Space-time continuum, any mathematical model that combines space and time into a single continuum * Continuum theory of specific heats of solids, see Deby ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Gent (hyperelastic Model)
The Alan N. Gent, 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 mathematical singularity, 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 J_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[1 - (I_1-3)x\right] ~;~~ x := \cfrac A Taylor series expansion of \ln\left[1 - (I_1-3)x\right] around x = 0 and taking the limit as x\rightarrow 0 leads to : W = \cfrac (I_1-3 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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 beta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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 model is a phenomenological model for the deformation of nearly incompressible, nonlinear Elasticity (physics), 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. The Yeoh model for incompressible rubber is a function only of I_1. For compressible rubbers, a dependence on I_3 is added on. Since a polynomial form of the strain energy density function is used but all the three invariants of the left Cauchy-Green deformation tensor are not, the Yeoh model is also called the reduced polynomial model. Yeoh model for incompressible rubbers Strain energy density functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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 A strain energy density function or stored energy density function is a scalar (mathematics), scalar-valued function (mathematics), function that relates the strain energy density of a material to the deformation gradient. : W = \hat(\boldsy ..., from which the stress–strain relationships can b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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 \bol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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 \frac(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 w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |