Slide_chpt02Mechanics

σyy
σxy σyx σxz+dσxz σxz σyz σxy+dσxy σxx+dσxx σzx σ
zy
dz
dx
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dy σzz
Finite Element Method by G. R. Liu and S. S. Quek
Dynamic equilibrium equations Hence, equilibrium equation in x direction
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Stress and strain Stresses at a point in a 3D solid:
σ xz = σ zx
σ zy = σ yz σ xy = σ yx
T
σyy σyz σyx
z
σzz σzx σxy σxz σxz σxy σzx σzz σzy σxx σyz σyy σyx
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Dynamic equilibrium equations Equilibrium of forces in x direction including the inertia forces
(σ xx + dσ xx )dydz − σ xx dydz + (σ yx + dσ yx )dxdz − σ yx dxdz + (σ zx + dσ zx )dxdy − σ zx dxdy + fx
sy.
c11 =
Eν E(1−ν ) , c12 = (1− 2 )(1+ν ) ν (1− 2 )(1+ν ) ν
G= E 2(1+ν )
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c11 − c12 =G , 2
Finite Element Method by G. R. Liu and S. S. Quek
Dynamic equilibrium equations Consider stresses on an infinitely small block σ +dσ
EQUATIONS FOR 3D SOLIDS EQUATIONS FOR 2D SOLIDS EQUATIONS FOR TRUSS MEMBERS EQUATIONS FOR BEAMS EQUATIONS FOR PLATES
Finite Element Method by G. R. Liu and S. S. Quek
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Stress and strain Strains in matrix form
ε =LU
where
0 0 ∂ ∂x 0 ∂ ∂y 0 0 0 ∂ ∂z L= ∂ ∂z ∂ ∂y 0 ∂ ∂z ∂ ∂x 0 0 ∂ ∂y ∂ ∂x
Finite Element Method by G. R. Liu and S. S. Quek
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Isotropy and anisotropy Anisotropic: the material property varies with direction. Composite materials: anisotropic, many material constants. Isotropic material: property is not direction dependent, two independent material constants.
Truss member
Beam member
Finite Element Method by G. R. Liu and S. S. Quek
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Different structural components
z
Plate and shell structures
y x
Neutral surface h Neutral surface
c14 c15 c16 εxx ε c24 c25 c26 yy c34 c35 c36 εzz ε c44 c45 c46 yz c55 c56 εxz c66 εxy
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Finite Element Method by G. R. Liu and S. S. Quek
Finite Element Method by G. R. Liu and S. S. Quek
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z
Different structural components
Truss and beam structures
y z x z fx
Sd
fs2 Sf V
fs1 Sf
fb1
fb2
y
Sd
Sd
x
fy2 fy1 x
Finite Element Method by G. R. Liu and S. S. Quek
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Statics and dynamics
Forces can be static and/or dynamic. Statics deals with the mechanics of solids and structures subjected to static loads. Dynamics deals with the mechanics of solids and structures subjected to dynamic loads. As statics is a special case of dynamics, the equations for statics can be derived by simply dropping out the dynamic terms in the dynamic equations.
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Finite Element Method by G. R. Liu and S. S. Quek
Boundary conditions Displacement (essential) boundary conditions Force (natural) boundary conditions
y
σxx σzy
σ = {σxx σyy σzz σyz σxz σxy }
x
Finite Element Method by G. R. Liu and S. S. Quek
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Stress and strain Strains
ε = {ε xx ε yy ε zz ε yz ε xz ε xy }
Stress and strain Constitutive equations Dynamic and static equilibrium equations
z
Sd
fs2 Sf V
fs1 Sf
fb1
fb2
y
Sd
Sd
x
Finite Element Method by G. R. Liu and S. S. Quek
∂σxx ∂σ yx ∂σzx ɺɺ + + + fx = ρu ∂x ∂y ∂z Equilibrium equations in y and z directions ∂σxy ∂σ yy ∂σzy ɺɺ + + + f y = ρv ∂x ∂y ∂z ∂σxz ∂σ yz ∂σzz ɺɺ + + + fz = ρw ∂x ∂y ∂z
Finite Element Method by G. R. Liu and S. S. Quek
u U = v w
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Constitutive equations σ=cε
or
σ xx c11 c12 c13 σ c22 c23 yy σ zz c33 = σ yz σ xz sy. σ xy
zz zz
σzx+dσzx σxy σyy
σzy+dσzy σxx σyz+dσyz σyy+dσyy σyx+dσyx
z dz
σyz
σyx σxz+dσxz σxx+dσxx σzy
σxz σxy+dσxy σzx σzz
dx
dy y x
Finite Element Method by G. R. Liu and S. S. Quek
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Finite Element Method by G. R. Liu and S. S. Quek
Elasticity and plasticity Elastic: the deformation in the solids disappears fully if it is unloaded. Plastic: the deformation in the solids cannot be fully recovered when it is unloaded. Elasticity deals with solids and structures of elastic materials. Plasticity deals with solids and structures of plastic materials.
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CONTENTS
INTRODUCTION
– – – – – Statics and dynamics Elasticity and plasticity Isotropy and anisotropy Boundary conditions Different structural components
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INTRODUCTION
Solids and structures are stressed when they are subjected to loads or forces. The stresses are, in general, not uniform as the forces usually vary with coordinates. The stresses lead to strains, which can be observed as a deformation or displacement. Solid mechanics and structural mechanics