In isotropic media, the stiffness tensor gives the relationship between the stresses resulting internal stresses and the strains resulting deformations.
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For an isotropic medium, the stiffness tensor has no preferred direction: an applied force will give the same displacements relative to the direction of the force no matter the direction in which the force is applied. In the isotropic case, the stiffness tensor may be written:. If the medium is inhomogeneous, the isotropic model is sensible if either the medium is piecewise-constant or weakly inhomogeneous; in the strongly inhomogeneous smooth model, anisotropy has to be accounted for.
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If the medium is homogeneous , then the elastic moduli will be independent of the position in the medium. The constitutive equation may now be written as:. This expression separates the stress into a scalar part on the left which may be associated with a scalar pressure, and a traceless part on the right which may be associated with shear forces.
A simpler expression is: . Since the constitutive equation is simply a set of linear equations, the strain may be expressed as a function of the stresses as: . Elastostatics is the study of linear elasticity under the conditions of equilibrium, in which all forces on the elastic body sum to zero, and the displacements are not a function of time. The equilibrium equations are then. In this case, the displacements are prescribed everywhere in the boundary.
In this approach, the strains and stresses are eliminated from the formulation, leaving the displacements as the unknowns to be solved for in the governing equations.
First, the strain-displacement equations are substituted into the constitutive equations Hooke's Law , eliminating the strains as unknowns:. In this way, the only unknowns left are the displacements, hence the name for this formulation.
The governing equations obtained in this manner are called the elastostatic equations , the special case of the Navier-Cauchy equations given below. These last 3 equations are the Navier-Cauchy equations, which can be also expressed in vector notation as. Once the displacement field has been calculated, the displacements can be replaced into the strain-displacement equations to solve for strains, which later are used in the constitutive equations to solve for stresses.
Noting that summed indices need not match, and that the partial derivatives commute, the two differential terms are seen to be the same and we have:. From the divergence equation, the first term on the left is zero Note: again, the summed indices need not match and we have:. In this case, the surface tractions are prescribed everywhere on the surface boundary. In this approach, the strains and displacements are eliminated leaving the stresses as the unknowns to be solved for in the governing equations.
Once the stress field is found, the strains are then found using the constitutive equations. There are six independent components of the stress tensor which need to be determined, yet in the displacement formulation, there are only three components of the displacement vector which need to be determined. This means that there are some constraints which must be placed upon the stress tensor, to reduce the number of degrees of freedom to three. Using the constitutive equations, these constraints are derived directly from corresponding constraints which must hold for the strain tensor, which also has six independent components.
The constraints on the strain tensor are derivable directly from the definition of the strain tensor as a function of the displacement vector field, which means that these constraints introduce no new concepts or information. It is the constraints on the strain tensor that are most easily understood. If the elastic medium is visualized as a set of infinitesimal cubes in the unstrained state, then after the medium is strained, an arbitrary strain tensor must yield a situation in which the distorted cubes still fit together without overlapping.
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In other words, for a given strain, there must exist a continuous vector field the displacement from which that strain tensor can be derived. The constraints on the strain tensor that are required to assure that this is the case were discovered by Saint Venant, and are called the " Saint Venant compatibility equations ".
These are 81 equations, 6 of which are independent non-trivial equations, which relate the different strain components. These are expressed in index notation as:. The strains in this equation are then expressed in terms of the stresses using the constitutive equations, which yields the corresponding constraints on the stress tensor. These constraints on the stress tensor are known as the Beltrami-Michell equations of compatibility:. These constraints, along with the equilibrium equation or equation of motion for elastodynamics allow the calculation of the stress tensor field.
Once the stress field has been calculated from these equations, the strains can be obtained from the constitutive equations, and the displacement field from the strain-displacement equations. An alternative solution technique is to express the stress tensor in terms of stress functions which automatically yield a solution to the equilibrium equation. The stress functions then obey a single differential equation which corresponds to the compatibility equations. Elastodynamics is the study of elastic waves and involves linear elasticity with variation in time.
An elastic wave is a type of mechanical wave that propagates in elastic or viscoelastic materials. The elasticity of the material provides the restoring force of the wave. When they occur in the Earth as the result of an earthquake or other disturbance, elastic waves are usually called seismic waves. The linear momentum equation is simply the equilibrium equation with an additional inertial term:. If the material is governed by anisotropic Hooke's law with the stiffness tensor homogeneous throughout the material , one obtains the displacement equation of elastodynamics :.
In isotropic media, the stiffness tensor has the form. If the material is homogeneous i. For plane waves , the above differential operator becomes the acoustic algebraic operator :. The associated waves are called longitudinal and shear elastic waves. In the seismological literature, the corresponding plane waves are called P-waves and S-waves see Seismic wave.
Elimination of displacements and strains from the governing equations leads to the Ignaczak equation of elastodynamics . The principal characteristics of this formulation include: 1 avoids gradients of compliance but introduces gradients of mass density; 2 it is derivable from a variational principle; 3 it is advantageous for handling traction initial-boundary value problems, 4 allows a tensorial classification of elastic waves, 5 offers a range of applications in elastic wave propagation problems; 6 can be extended to dynamics of classical or micropolar solids with interacting fields of diverse types thermoelastic, fluid-saturated porous, piezoelectro-elastic Subjects Primary: 74B Nonlinear elasticity 35L Nonlinear second-order hyperbolic equations 35L Initial-boundary value problems for second-order hyperbolic systems 74H Existence of solutions 35A Existence problems: global existence, local existence, non-existence 35D Weak solutions 35K Quasilinear parabolic equations with p-Laplacian.
Almost global existence of weak solutions for the nonlinear elastodynamics system for a class of strain energies. Differential Equations 23 , no. Read more about accessing full-text Buy article. Article information Source Adv. Export citation. Export Cancel. Khayyam Publishing, Inc. Editorial Board For Authors Subscriptions. You have access to this content. You have partial access to this content.