Reissner-Mindlin and Kirchhoff-Love plates

8.1. Reissner-Mindlin and Kirchhoff-Love plates#

In this section we quickly derive the Reissner-Mindlin and Kirchhoff-Love plate equations starting from a thin 3D plate linear elasticity problem: Find u:ΩR3 such that for all test functions v

ΩC(ε(u)):ε(v)dx=Ωfvdx,

with ε(u) the symmetric gradient of u and the elasticity tensor

Cε=E(1+ν)(12ν)((12ν)ε+νtr(ε)I3×3),

where E and ν denote the Young’s modulus and Poisson ratio, respectively.

8.1.1. Reissner-Mindlin plate#

Assuming a thin three-dimensional plate Ω=(0,1)2×(t/2,t/2) we can perform a dimension reduction to the mid-surface S=(0,1)2×{0}. We assume the structure to be clamped on all boundaries except the top and bottom one, where homogeneous Neumann boundary conditions are prescribed. We postulate kinematic assumptions, which are named after Reissner and Mindlin:

  1. Lines normal to the mid-surface get deformed linearly, they remain lines.

  2. The displacements in z-direction are independent of the z-coordinate.

  3. Points on the mid-surface can only be deformed in z-direction.

  4. Stresses σ33 in normal direction vanish (called plane-stress assumption).

With H1-H3 the displacements are of the form

u1(x,y,z)=zβ1(x,y),u2(x,y,z)=zβ2(x,y),u3(x,y,z)=w(x,y).

Hypothesis H3 directly enforces that the horizontal displacements of the membrane problem is eliminated and yields together with H1 the form of the shearing related horizontal displacements u1 and u2. H2 in combination with H1 gives the form of the vertical displacement. Assumption H4 is needed as from u3=w(x,y) there follows ε33(u)=0, i.e., no strains in thickness direction. Using the stress-strain relation to compute σ33 yields σ33=E(1+ν)(12ν)((1ν)ε33+ν(ε11+ε22))=λ(ε11+ε22)0 in general. This is non-physical and leads to a not asymptotically correct model. It induces a too stiff behavior yielding artificial stiffness. Therefore σ33=0 is postulated to re-obtain an asymptotically correct model, which converges to the 3D solution in the limit of vanishing thickness. Setting σ33=0 induces that the material can stretch in thickness direction without inducing stresses.

The 3D elasticity strain tensor reads

ε(u)=12(u+u)=(z1β1z12(2β1+1β2)12(1wβ1)z2β212(2wβ2)sym0).

From H4, σ33=0, we can express ε33=ν1ν(ε11+ε22) by using the stress-strain relation σ=Cε

(σ11σ22σ33σ12σ13σ23)=E(1+ν)(12ν)(1ννν0ν1νννν1ν12ν12ν012ν)(ε11ε22ε33ε12ε13ε23),

and reinserting yields

(σ11σ22σ12σ13σ23)=E1ν2(1ν0ν11ν1ν01ν)(ε11ε22ε12ε13ε23).

For example, there holds

σ11=E(1+ν)(12ν)((1ν)ε11+νε22+νε33)=E(1+ν)(12ν)((1νν21ν)ε11+(νν21ν)ε22)=E(1ν2)(12ν)((12ν+ν2ν2)ε11+(ν2ν2)ε22)=E1ν2(ε11+νε22).

The energy εC2=Cε:ε=σ:ε reads

σ:ε=i,j=12σijεij+2j=12ε3jσ3j=E1+ν(i,j=12εij2+ν1ν(ε11+ε22)2+2j=12ε3j2).

Next, we integrate the arising terms over the thickness and insert the strain definition

t/2t/2i,j=12εij2dz=t/2t/2z2ε(β):ε(β)dz=t312ε(β):ε(β),t/2t/2(ε11+ε22)2dz=t312div(β)2,t/2t/22j=12ε3j2dz=t/2t/224(wβ)(wβ)dz=t2wβ2,

where , ε, and div denote the differential operators acting only on the first two indices i,j=1,2.

Thus, the total energy W(w,β)=12Ωσ:εd(x,y,z) becomes with the notation ds=d(x,y)

W(w,β)=12Sσ:εd(s,z)=t3E24(1ν2)S(1ν)ε(β)2+νdiv(β)2ds+Gt2Swβ2ds,

where G=E2(1+ν) denotes the shearing modulus. Classically, the shear correction factor κ=56 is additionally inserted into the shearing energy term to compensate high-order effects of the shear stresses, which are not constant through the thickness (in some variational methods to derive plate equations the factor 5/6 directly appears).

The right-hand side f is assumed to act only vertically on the plate and is independent of the thickness f=(0,0,fz(x,y)). Integrating over the thickness and rescaling f:=t2fz leads to the term t3Sfvds.

To simplify notation we set Ω=S, use dx for integration over the mid-surface, neglect the underline for the differential operators, and define the plate elasticity tensor

DA:=E1ν2((1ν)A+νtr(A)I).

All together, taking the variations of the Reissner-Mindlin plate energy (with the shear correction factor κ) and dividing by t3 yields the clamped Reissner-Mindlin plate equation: Find (w,β)H01(Ω)×[H01(Ω)]2 such that for all (v,δ)H01(Ω)×[H01(Ω)]2

112ΩDε(β):ε(δ)dx+κGt2Ω(wβ)(vδ)dx=Ωfvdx.

Depending on the different combination of Dirichlet (D) and Neumann (N) boundary conditions for the vertical deflection w and the rotations β we distinguish between the following plate boundary conditions. Let M:=112Dε(β) be the bending moment tensor and Q:=κGt2(wβ) the shear force.

boundary conditions

w

βn

βt

resulting conditions

clamped

D

D

D

w=0, β=0

free

N

N

N

Qn=0, Mn=0

hard simply supported

D

N

D

w=0, nMn=0, βt=0

soft simply supported

D

N

N

w=0, Mn=0

8.1.2. Kirchhoff-Love plate#

For thin plates like metal sheets the Kirchhoff-Love hypothesis is assumed additionally to H1-H4.

  1. Lines normal to the mid-surface are after deformation again normal to the deformed mid-surface.

It states that normal vectors of the original plate stay perpendicular to the mid-surface of the deformed plate. This means that no shearing occurs, i.e., the rotations β from the Reissner-Mindlin plate equation can be eliminated by the gradient of the vertical deflection w of the mid-surface, β=w. Note, that w is the linearization of the rotated normal vector of the plate.

AngleRM AngleKL

Eliminating β from the Reissner-Mindlin plate equation leads to the Kirchhoff-Love plate equation, which is a fourth order problem. Assuming clamped boundary conditions it reads: Find wH02(Ω) such that for all vH02(Ω)

ΩD2w:2vdx=Ωfvdx.

The thickness parameter t does not enter the equation. The problem is well-posed for wH02(Ω) and reads in strong form

div(div(D2w))=f in Ω,w=wn=0 on Ω.

Additionally to clamped boundary conditions also free and simply supported boundary conditions can be prescribed. The general case in strong form reads:

div(div(σ))=f,σ:=D2w in Ω,w=0,wn=0 on Γc,w=0,σnn=0 on Γs,σnn=0,σntt+div(σ)n=0 on Γf,[[σnt]]x=σn1t1(x)σn2t2(x)=0xVΓf,

where the boundary Γ=Ω splits into clamped, simply supported, and free boundaries Γc, Γs, and Γf, respectively. VΓf denotes the set of corner points where the two adjacent edges belong to Γf. Here, n and t denote the outer normal and tangential vector on the plate boundary. Physically, σnn:=nσn is the normal bending moment, t(tσn)+ndiv(σ) the effective transverse shear force, and σnt:=tσn the torsion moment. Further, the shear force Q is given by Q=div(σ).