Linear elasticity

Linear elasticity is a mathematical model of how solid objects deform and become internally stressed due to prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mechanics. ∂ τ x y ∂ x + ∂ σ y ∂ y + ∂ τ z y ∂ z + F y = ρ ∂ 2 u y ∂ t 2 {displaystyle {frac {partial au _{xy}}{partial x}}+{frac {partial sigma _{y}}{partial y}}+{frac {partial au _{zy}}{partial z}}+F_{y}= ho {frac {partial ^{2}u_{y}}{partial t^{2}}},!} ∂ τ x y ∂ x + ∂ σ y ∂ y + ∂ τ z y ∂ z + F y = 0 {displaystyle {frac {partial au _{xy}}{partial x}}+{frac {partial sigma _{y}}{partial y}}+{frac {partial au _{zy}}{partial z}}+F_{y}=0,!} Then substituting these equations into the equilibrium equation in the x {displaystyle x,!} -direction we have ∂ 2 ϵ y ∂ z 2 + ∂ 2 ϵ z ∂ y 2 = 2 ∂ 2 ϵ y z ∂ y ∂ z {displaystyle {frac {partial ^{2}epsilon _{y}}{partial z^{2}}}+{frac {partial ^{2}epsilon _{z}}{partial y^{2}}}=2{frac {partial ^{2}epsilon _{yz}}{partial ypartial z}},!} where ν {displaystyle u ,!} is Poisson's ratio, the solution may be expressed as Linear elasticity is a mathematical model of how solid objects deform and become internally stressed due to prescribed loading conditions. It is a simplification of the more general nonlinear theory of elasticity and a branch of continuum mechanics. The fundamental 'linearizing' assumptions of linear elasticity are: infinitesimal strains or 'small' deformations (or strains) and linear relationships between the components of stress and strain. In addition linear elasticity is valid only for stress states that do not produce yielding. These assumptions are reasonable for many engineering materials and engineering design scenarios. Linear elasticity is therefore used extensively in structural analysis and engineering design, often with the aid of finite element analysis. 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: where σ {displaystyle {oldsymbol {sigma }}} is the Cauchy stress tensor, ε {displaystyle {oldsymbol {varepsilon }}} is the infinitesimal strain tensor, u {displaystyle mathbf {u} } is the displacement vector, C {displaystyle {mathsf {C}}} is the fourth-order stiffness tensor, F {displaystyle mathbf {F} } is the body force per unit volume, ρ {displaystyle ho } is the mass density, ∇ {displaystyle {oldsymbol { abla }}} represents the nabla operator, ( ∙ ) T {displaystyle (ullet )^{mathrm {T} }} represents a transpose, ( ∙ ) ¨ {displaystyle {ddot {(ullet )}}} represents the second derivative with respect to time, and A : B = A i j B i j {displaystyle {mathsf {A}}:{mathsf {B}}=A_{ij}B_{ij}} is the inner product of two second-order tensors (summation over repeated indices is implied). Expressed in terms of components with respect to a rectangular Cartesian coordinate system, the governing equations of linear elasticity are: An elastostatic boundary value problem for an isotropic-homogeneous media is a system of 15 independent equations and equal number of unknowns (3 equilibrium equations, 6 strain-displacement equations, and 6 constitutive equations). Specifying the boundary conditions, the boundary value problem is completely defined. To solve the system two approaches can be taken according to boundary conditions of the boundary value problem: a displacement formulation, and a stress formulation. In cylindrical coordinates ( r , θ , z {displaystyle r, heta ,z} ) the equations of motion are