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Hypoelastic Material
In continuum mechanics, a hypoelastic material is an elastic material that has a constitutive model independent of finite strain measures except in the linearized case. Hypoelastic material models are distinct from hyperelastic material models (or standard elasticity models) in that, except under special circumstances, they cannot be derived from a strain energy density function. Overview A hypoelastic material can be rigorously defined as one that is modeled using a constitutive equation satisfying the following two criteria: # The Cauchy stress \boldsymbol at time t depends only on the order in which the body has occupied its past configurations, but not on the time rate at which these past configurations were traversed. As a special case, this criterion includes a Cauchy elastic material, for which the current stress depends only on the current configuration rather than the history of past configurations. # There is a tensor-valued function G such that \dot = G(\boldsymbol,\bo ... [...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] |
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 transfer, fluid flow, mass transport, and electromagnetic potential. The FEM is a general numerical method for solving partial differential equations in two or three space variables (i.e., some boundary value problems). To solve a problem, the FEM subdivides a large system into smaller, simpler parts that are called finite elements. This is achieved by a particular space discretization in the space dimensions, which is implemented by the construction of a mesh of the object: the numerical domain for the solution, which has a finite number of points. The finite element method formulation of a boundary value problem finally results in a system of algebraic equations. The method approximates the unknown function over the domain. The simple ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infinitesimal Strain Theory
In continuum mechanics, the infinitesimal strain theory is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller (indeed, infinitesimally smaller) than any relevant dimension of the body; so that its geometry and the constitutive properties of the material (such as density and stiffness) at each point of space can be assumed to be unchanged by the deformation. With this assumption, the equations of continuum mechanics are considerably simplified. This approach may also be called small deformation theory, small displacement theory, or small displacement-gradient theory. It is contrasted with the finite strain theory where the opposite assumption is made. The infinitesimal strain theory is commonly adopted in civil and mechanical engineering for the stress analysis of structures built from relatively stiff elastic materials like concrete and steel, since a common goal in t ... [...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] |
Principle Of Material Objectivity
Walter Noll (January 7, 1925 June 6, 2017) was a mathematician, and Professor Emeritus at Carnegie Mellon University. He is best known for developing mathematical tools of classical mechanics, thermodynamics, and continuum mechanics. Biography Born in Berlin, Germany, Noll had his school education in a suburb of Berlin. In 1954, Noll earned a Ph.D. in Applied Mathematics from Indiana University under Clifford Truesdell. His thesis "On the Continuity of the Solid and Fluid States" was published both in '' Journal of Rational Mechanics and Analysis'' and in one of Truesdell's books. Noll thanks Jerald Ericksen for his critical input to the thesis. Noll has served as a visiting professor at the Johns Hopkins University, the University of Karlsruhe, the Israel Institute of Technology, the Institut National Polytechnique de Lorraine in Nancy, the University of Pisa, the University of Pavia, and the University of Oxford. In 2012 he became a fellow of the American Mathematical Socie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Objective Stress Rates
300px, Predictions from three objective stress rates under shear In continuum mechanics, objective stress rates are time derivatives of stress that do not depend on the frame of reference. Many constitutive equations are designed in the form of a relation between a stress-rate and a strain-rate (or the rate of deformation tensor). The mechanical response of a material should not depend on the frame of reference. In other words, material constitutive equations should be frame-indifferent (objective). If the stress and strain measures are material quantities then objectivity is automatically satisfied. However, if the quantities are spatial, then the objectivity of the stress-rate is not guaranteed even if the strain-rate is objective. There are numerous objective stress rates in continuum mechanics – all of which can be shown to be special forms of Lie derivatives. Some of the widely used objective stress rates are: # the Truesdell rate of the Cauchy stress tensor, # the ... [...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] |
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] |
Objective Stress Rates
300px, Predictions from three objective stress rates under shear In continuum mechanics, objective stress rates are time derivatives of stress that do not depend on the frame of reference. Many constitutive equations are designed in the form of a relation between a stress-rate and a strain-rate (or the rate of deformation tensor). The mechanical response of a material should not depend on the frame of reference. In other words, material constitutive equations should be frame-indifferent (objective). If the stress and strain measures are material quantities then objectivity is automatically satisfied. However, if the quantities are spatial, then the objectivity of the stress-rate is not guaranteed even if the strain-rate is objective. There are numerous objective stress rates in continuum mechanics – all of which can be shown to be special forms of Lie derivatives. Some of the widely used objective stress rates are: # the Truesdell rate of the Cauchy stress tensor, # the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy Stress Tensor
In continuum mechanics, the Cauchy stress tensor \boldsymbol\sigma, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components \sigma_ that completely define the state of stress at a point inside a material in the deformed state, placement, or configuration. The tensor relates a unit-length direction vector e to the traction vector T(e) across an imaginary surface perpendicular to e: :\mathbf^ = \mathbf e \cdot\boldsymbol\quad \text \quad T_^= \sigma_e_i, or, :\leftright\leftrightcdot \leftright The SI units of both stress tensor and traction vector are N/m2, corresponding to the stress scalar. The unit vector is dimensionless. The Cauchy stress tensor obeys the tensor transformation law under a change in the system of coordinates. A graphical representation of this transformation law is the Mohr's circle for stress. The Cauchy stress tensor is used for stress analysis of materi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Material Objectivity
Walter Noll (January 7, 1925 June 6, 2017) was a mathematician, and Professor Emeritus at Carnegie Mellon University. He is best known for developing mathematical tools of classical mechanics, thermodynamics, and continuum mechanics. Biography Born in Berlin, Germany, Noll had his school education in a suburb of Berlin. In 1954, Noll earned a Ph.D. in Applied Mathematics from Indiana University under Clifford Truesdell. His thesis "On the Continuity of the Solid and Fluid States" was published both in '' Journal of Rational Mechanics and Analysis'' and in one of Truesdell's books. Noll thanks Jerald Ericksen for his critical input to the thesis. Noll has served as a visiting professor at the Johns Hopkins University, the University of Karlsruhe, the Israel Institute of Technology, the Institut National Polytechnique de Lorraine in Nancy, the University of Pisa, the University of Pavia, and the University of Oxford. In 2012 he became a fellow of the American Mathematical Socie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kirchhoff Stress
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 surface ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
