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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] |
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] |
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] |
Non-Newtonian Fluids
A non-Newtonian fluid is a fluid that does not follow Newton's law of viscosity, i.e., constant viscosity independent of stress. In non-Newtonian fluids, viscosity can change when under force to either more liquid or more solid. Ketchup, for example, becomes runnier when shaken and is thus a non-Newtonian fluid. Many salt solutions and molten polymers are non-Newtonian fluids, as are many commonly found substances such as custard, toothpaste, starch suspensions, corn starch, paint, blood, melted butter, and shampoo. Most commonly, the viscosity (the gradual deformation by shear or tensile stresses) of non-Newtonian fluids is dependent on shear rate or shear rate history. Some non-Newtonian fluids with shear-independent viscosity, however, still exhibit normal stress-differences or other non-Newtonian behavior. In a Newtonian fluid, the relation between the shear stress and the shear rate is linear, passing through the origin, the constant of proportionality being the coeffic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elasticity (physics)
In physics and materials science, elasticity is the ability of a 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 deform when adequate 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'', 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 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 rubbers and other polymers, elasticity is caused by the stretching of polymer chains when forces are applied. Hooke's law states that the force required to deform elastic objects should be directly proportional to the distance of ... [...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] |
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] |
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] |
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] |
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] |
Simple Shear
Simple shear is a deformation in which parallel planes in a material remain parallel and maintain a constant distance, while translating relative to each other. In fluid mechanics In fluid mechanics, simple shear is a special case of deformation where only one component of velocity vectors has a non-zero value: :V_x=f(x,y) :V_y=V_z=0 And the gradient of velocity is constant and perpendicular to the velocity itself: :\frac = \dot \gamma , where \dot \gamma is the shear rate and: :\frac = \frac = 0 The displacement gradient tensor Γ for this deformation has only one nonzero term: :\Gamma = \begin 0 & & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end Simple shear with the rate \dot \gamma is the combination of pure shear strain with the rate of \dot \gamma and rotation with the rate of \dot \gamma: :\Gamma = \begin \underbrace \begin 0 & & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end \\ \mbox\end = \begin \underbrace \begin 0 & & 0 \\ & 0 & 0 \\ 0 & 0 & 0 \end \\ \mbox \end + \begi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stress (physics)
In continuum mechanics, stress is a physical quantity. It is a quantity that describes the magnitude of forces that cause deformation. Stress is defined as ''force per unit area''. When an object is pulled apart by a force it will cause elongation which is also known as deformation, like the stretching of an elastic band, it is called tensile stress. But, when the forces result in the compression of an object, it is called compressive stress. It results when forces like Tension (physics), tension or Compression (physics), compression act on a body. The greater this force and the smaller the cross-sectional area of the body on which it acts, the greater the stress. Therefore, stress is measured in newton per square meter (N/m2) or pascal (Pa). Stress expresses the internal forces that neighbouring particles of a continuous material exert on each other, while deformation (mechanics)#Strain, strain is the measure of the deformation of the material. For example, when a solid vertic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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