Linear elasticity
Linear elasticity is a mathematical model as to how solid objects deform and become internally stressed by prescribed loading conditions.
It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mechanics.
In addition linear elasticity is valid only for stress states that do not produce yielding.
Equations governing a linear elastic boundary value problem are based on three tensor partial differential equations for the balance of linear momentum and six infinitesimal strain-displacement relations.
The system of differential equations is completed by a set of linear algebraic constitutive relations.
In direct tensor form that is independent of the choice of coordinate system, these governing equations are:[1] where
is the inner product of two second-order tensors (summation over repeated indices is implied).
and the constitutive relations are the same as in Cartesian coordinates, except that the indices 1,2,3 now stand for
In the isotropic case, the stiffness tensor may be written:[citation needed]
In engineering notation (with tau as shear stress), This section will discuss only the isotropic homogeneous case.
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.
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:
Taking the Laplacian of both sides of the elastostatic equation, and assuming in addition
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 approach, the strains and displacements are eliminated leaving the stresses as the unknowns to be solved for in the governing equations.
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.
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.
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.
In the special situation where the body force is homogeneous, the above equations reduce to[6]
The most important solution of the Navier–Cauchy or elastostatic equation is for that of a force acting at a point in an infinite isotropic medium.
It is particularly helpful to write the displacement in cylindrical coordinates for a point force
Another useful solution is that of a point force acting on the surface of an infinite half-space.
{\displaystyle G_{ik}={\frac {1}{4\pi \mu }}{\begin{bmatrix}{\frac {b}{r}}+{\frac {x^{2}}{r^{3}}}-{\frac {ax^{2}}{r(r+z)^{2}}}-{\frac {az}{r(r+z)}}&{\frac {xy}{r^{3}}}-{\frac {axy}{r(r+z)^{2}}}&{\frac {xz}{r^{3}}}-{\frac {ax}{r(r+z)}}\\{\frac {yx}{r^{3}}}-{\frac {ayx}{r(r+z)^{2}}}&{\frac {b}{r}}+{\frac {y^{2}}{r^{3}}}-{\frac {ay^{2}}{r(r+z)^{2}}}-{\frac {az}{r(r+z)}}&{\frac {yz}{r^{3}}}-{\frac {ay}{r(r+z)}}\\{\frac {zx}{r^{3}}}-{\frac {ax}{r(r+z)}}&{\frac {zy}{r^{3}}}-{\frac {ay}{r(r+z)}}&{\frac {b}{r}}+{\frac {z^{2}}{r^{3}}}\end{bmatrix}}}
If the material is isotropic and homogeneous, one obtains the (general, or transient) Navier–Cauchy equation:
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...) as well as nonlinear media.
Voigt notation is the standard mapping for tensor indices,
is symmetric, this is a result of the existence of a strain energy density function which satisfies
The simplest anisotropic case, that of cubic symmetry has 3 independent elements:
The case of transverse isotropy, also called polar anisotropy, (with a single axis (the 3-axis) of symmetry) has 5 independent elements:
The case of orthotropy (the symmetry of a brick) has 9 independent elements:
