Kirchhoff–Love plate theory

The Kirchhoff–Love theory of plates is a two-dimensional mathematical model that is used to determine the stresses and deformations in thin plates subjected to forces and moments. This theory is an extension of Euler-Bernoulli beam theory and was developed in 1888 by Love using assumptions proposed by Kirchhoff. The theory assumes that a mid-surface plane can be used to represent a three-dimensional plate in two-dimensional form. u α ( x ) = u α 0 ( x 1 , x 2 ) − x 3   ∂ w 0 ∂ x α ≡ u α 0 − x 3   w , α 0   ;     α = 1 , 2 u 3 ( x ) = w 0 ( x 1 , x 2 ) {displaystyle {egin{aligned}u_{alpha }(mathbf {x} )&=u_{alpha }^{0}(x_{1},x_{2})-x_{3}~{frac {partial w^{0}}{partial x_{alpha }}}equiv u_{alpha }^{0}-x_{3}~w_{,alpha }^{0}~;~~alpha =1,2\u_{3}(mathbf {x} )&=w^{0}(x_{1},x_{2})end{aligned}}} ε α β = 1 2 ( u α , β 0 + u β , α 0 ) − x 3   w , α β 0 ε α 3 = − w , α 0 + w , α 0 = 0 ε 33 = 0 {displaystyle {egin{aligned}varepsilon _{alpha eta }&={ frac {1}{2}}(u_{alpha ,eta }^{0}+u_{eta ,alpha }^{0})-x_{3}~w_{,alpha eta }^{0}\varepsilon _{alpha 3}&=-w_{,alpha }^{0}+w_{,alpha }^{0}=0\varepsilon _{33}&=0end{aligned}}} N α β , α = 0 N α β := ∫ − h h σ α β   d x 3 M α β , α β − q = 0 M α β := ∫ − h h x 3   σ α β   d x 3 {displaystyle {egin{aligned}N_{alpha eta ,alpha }&=0quad quad N_{alpha eta }:=int _{-h}^{h}sigma _{alpha eta }~dx_{3}\M_{alpha eta ,alpha eta }-q&=0quad quad M_{alpha eta }:=int _{-h}^{h}x_{3}~sigma _{alpha eta }~dx_{3}end{aligned}}} where the thickness of the plate is 2 h {displaystyle 2h} and the stress resultants and stress moment resultants are defined as ∇ 2 ∇ 2 w = 0 {displaystyle abla ^{2} abla ^{2}w=0} and the stress-strain relations are ∇ 2 ∇ 2 w = q D   ;     D := 2 h 3 E 3 ( 1 − ν 2 ) {displaystyle abla ^{2} abla ^{2}w={cfrac {q}{D}}~;~~D:={cfrac {2h^{3}E}{3(1- u ^{2})}}} where q {displaystyle q} is a distributed transverse load (per unit area). Substitution of the expressions for the derivatives of M α β {displaystyle M_{alpha eta }} into the governing equation gives ∇ 2 ∇ 2 w = − q D . {displaystyle abla ^{2} abla ^{2}w=-{frac {q}{D}},.} N α β , β = J 1   u ¨ α 0 M α β , α β + q ( x , t ) = J 1   w ¨ 0 − J 3   w ¨ , α α 0 {displaystyle {egin{aligned}N_{alpha eta ,eta }&=J_{1}~{ddot {u}}_{alpha }^{0}\M_{alpha eta ,alpha eta }+q(x,t)&=J_{1}~{ddot {w}}^{0}-J_{3}~{ddot {w}}_{,alpha alpha }^{0}end{aligned}}} The total kinetic energy of the plate is given bymode k = 0, p = 1mode k = 0, p = 2mode k = 1, p = 2For an isotropic and homogeneous plate, the stress-strain relations are The Kirchhoff–Love theory of plates is a two-dimensional mathematical model that is used to determine the stresses and deformations in thin plates subjected to forces and moments. This theory is an extension of Euler-Bernoulli beam theory and was developed in 1888 by Love using assumptions proposed by Kirchhoff. The theory assumes that a mid-surface plane can be used to represent a three-dimensional plate in two-dimensional form. The following kinematic assumptions that are made in this theory: Let the position vector of a point in the undeformed plate be x {displaystyle mathbf {x} } . Then The vectors e i {displaystyle {oldsymbol {e}}_{i}} form a Cartesian basis with origin on the mid-surface of the plate, x 1 {displaystyle x_{1}} and x 2 {displaystyle x_{2}} are the Cartesian coordinates on the mid-surface of the undeformed plate, and x 3 {displaystyle x_{3}} is the coordinate for the thickness direction. Let the displacement of a point in the plate be u ( x ) {displaystyle mathbf {u} (mathbf {x} )} . Then This displacement can be decomposed into a vector sum of the mid-surface displacement u α 0 {displaystyle u_{alpha }^{0}} and an out-of-plane displacement w 0 {displaystyle w^{0}} in the x 3 {displaystyle x_{3}} direction. We can write the in-plane displacement of the mid-surface as Note that the index α {displaystyle alpha } takes the values 1 and 2 but not 3.