Introduction
Plane elasticity
Three-dimensional elasticity
Summary
Linearized theory of elasticity Arie Verhoeven
[email protected]
CASA Seminar, May 24, 2006
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Introduction
Plane elasticity
Three-dimensional elasticity
Summary
Seminar: Continuum mechanics 1 2 3 4 5 6 7 8 9
Stress and stress principles Strain and deformation General principles Constitutive equations Fluid mechanics Linearized theory of elasticity Complex representation of the stress function Principles of virtual work The boundary element method
Bart Nowak Mark van Kraaij Ali Etaati Godwin Kakuba Peter in ’t Panhuis Arie Verhoeven Erwin Vondenhoff
March 8 March 29 April 12 April 19 May 3 May 24 June 7
Andriy Hlod Zoran Ilievsky
June 21 June 28
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Introduction
Plane elasticity
Three-dimensional elasticity
Summary
Outline 1
Introduction
2
Plane elasticity Rectangular coordinates Cylindrical and polar coordinates
3
Three-dimensional elasticity Direct solution of Navier’s equation Examples
4
Summary
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Introduction
Plane elasticity
Three-dimensional elasticity
Summary
Outline 1
Introduction
2
Plane elasticity Rectangular coordinates Cylindrical and polar coordinates
3
Three-dimensional elasticity Direct solution of Navier’s equation Examples
4
Summary
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Introduction
Plane elasticity
Three-dimensional elasticity
Summary
The Cauchy stress tensor For a linear elastic solid we have the identity: TJi (X, t) = T˜Ji (x + u(x, t)). In terms of Cartesian components, the first Piola-Kirchhoff stress tensor TJi0 is related to the Cauchy stress Tri at the points x = X + u by TJi0 =
ρ0 ∂XJ Tri . ρ ∂xr
(1)
The displacement-gradient components are small compared to unity. The equations of motion in the reference state are given by ∂TJi0 d 2 xi + ρ0 b0i = ρ0 2 . ∂XJ dt
(2)
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Introduction
Plane elasticity
Three-dimensional elasticity
Summary
Field equations Field Equations of Linearized Isotropic Isothermal Elasticity 3 6
Eqs. of Motion Hooke’s Law Eqs.
6
Geometric Eqs.
2
∂Tji ∂Xj
+ ρbi = ρ ∂∂tu2i Tij = λEkk δij + 2µE ij
Eij =
1 2
∂ui ∂Xj
+
∂uj ∂Xi
(3) (4) (5)
15 eqs. for 6 stresses, 6 strains, 3 displacements The two Lamé elastic constants λ and µ, introduced by Lamé in 1852, are related to the more familiar shear modulus G, Young’s modulus E, and Poisson’s ratio ν as follows: µ=G=
E 2(1 + ν)
and λ =
νE . (1 + ν)(1 − 2ν)
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Introduction
Plane elasticity
Three-dimensional elasticity
Summary
Boundary conditions 1
Displacement boundary conditions, with the three components ui prescribed on the boundary.
2
Traction boundary conditions, with the three traction components ti = Tji nj prescribed at a boundary point.
3
Mixed boundary conditions include cases where 1
2
Displacement boundary conditions are prescribed on a part of the bounding surface, while traction boundary conditions are prescribed on the remainder, or at each point of the boundary we choose local rectangular Cartesian axes X¯i and then prescribe: 1 2 3
¯1 or ¯t1 , but not both, u ¯2 or ¯t2 , but not both, and u ¯3 or ¯t3 , but not both. u
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Introduction
Plane elasticity
Three-dimensional elasticity
Summary
Navier’s displacement equations Equations of this form were given by Navier in a memoir of 1821, published in 1827, but they contained only one elastic constant because they were deduced from an inadequate molecular model. The two-constant version was given by Cauchy in 1822. Navier Equation 2 (λ + µ)∇(∇ · u) + µ∇2 u + ρb = ρ ∂∂tu 2
(8)
Traction Boundary Condition ¯ ·n ¯ ·n ˆ + µ(u∇ ˆ + ∇u) ˆ = prescribed function λ(∇ · u)n Elastostatics:
∂2u ∂t 2
(9)
= 0. /centre for analysis, scientific computing and applications
Introduction
Plane elasticity
Three-dimensional elasticity
Summary
Outline 1
Introduction
2
Plane elasticity Rectangular coordinates Cylindrical and polar coordinates
3
Three-dimensional elasticity Direct solution of Navier’s equation Examples
4
Summary
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Introduction
Plane elasticity
Three-dimensional elasticity
Summary
Outline 1
Introduction
2
Plane elasticity Rectangular coordinates Cylindrical and polar coordinates
3
Three-dimensional elasticity Direct solution of Navier’s equation Examples
4
Summary
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Introduction
Plane elasticity
Three-dimensional elasticity
Summary
Plane elasticity (1) In plane deformation, the assumptions uz = 0 and ux and uy indepenent of z lead to only three independent strain components ex , ey , and exy = 12 γxy , which are independent of z, a state of plane strain parallel to the xy -plane. The isotropic Hooke’s law reduces to σx = λe + 2Gex σy = λe + 2Gey (3) τxy = 2Gexy with, in addition, (4)
σz = λe = ν(σx + σy ), where e = ex + ey ,
G=
E , 2(1 + ν)
and λ =
Eν . (1 + ν)(1 − 2ν)
(5)
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Introduction
Plane elasticity
Three-dimensional elasticity
Summary
Plane elasticity (2)
To these must be added two equations of motion ) ∂τxy ∂σx ∂ 2 ux ∂x + ∂y + ρbx = ρ ∂t 2 ∂τxy ∂x
+
∂σy ∂y
+ +ρby = ρ
∂ 2 uy ∂t 2
(6)
and one compatibility equation ∂ 2 ey ∂ 2 exy ∂ 2 ex + =2 , 2 2 ∂y ∂x ∂x∂y
(7)
if for small displacements we ignore the difference between the material coordinates X , Y and the spatial coordinates x, y .
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Introduction
Plane elasticity
Three-dimensional elasticity
Summary
Particular solution for body forces The linearity can also be used to construct the solution in two parts: σij = σijH + σijP , eij = eijH + eijP . The particular solution σijP , eijP satisfies the given equations with given body-force distributions but not the boundary conditions. The distribution σijH , eijH satisfies the homogeneous differential equations (with no body force) and suitably modified boundary conditions. When the body force is simply the weight, say bx = 0, by = −g, then a possible particular solution is σyP = ρgy − C,
P σxP = τxy = 0,
(8)
where −C is the value of σyP at y = 0.
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Introduction
Plane elasticity
Three-dimensional elasticity
Summary
Airy stress function For plane strain with no body forces, the equilibrium equations are identically satisfied if the stresses are related to a scalar function φ(x, y ), called Airy’s stress function, by the equations σx =
∂2φ , ∂y 2
σy =
∂2φ , ∂x 2
τxy = −
∂2φ . ∂x∂y
(9)
The compatibility equation then becomes the biharmonic equation ∇21 (∇21 )φ = 0
(10)
or ∇41 φ = 0, where ∇1 =
∂2 ∂x 2
+
∂2 ∂y 2
is the 2D Laplace operator.
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Introduction
Plane elasticity
Three-dimensional elasticity
Summary
Boundary conditions for Airy stress function
Airy stress function useful for boundary conditions for tractions. Since ti = σji nj , we have tx = σx nx + τxy ny and ty = τxy nx + σy ny or tx
=
ty
=
∂ 2 φ dy + ∂y 2 ds ∂ 2 φ dy − ∂x∂y ds
∂ 2 φ dx ∂φ d ∂x∂y ds = ds ∂y 2 ∂φ ∂ φ dx d = − ds ∂x . ∂x 2 ds
) (11)
R Hence, integrating along the boundary, we obtain ∂φ = − C ty ds + C1 ∂x R dφ ∂φ dx ∂φ dy and ∂φ ∂y = C tx ds + C2 . Now we can calculate ds = ∂x ds + ∂y ds , R dφ ∂φ dy ∂φ dx dφ dn = ∂x ds − ∂y ds and φ = C ds ds + C3 at the boundary.
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Introduction
Plane elasticity
Three-dimensional elasticity
Summary
Outline 1
Introduction
2
Plane elasticity Rectangular coordinates Cylindrical and polar coordinates
3
Three-dimensional elasticity Direct solution of Navier’s equation Examples
4
Summary
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Introduction
Plane elasticity
Three-dimensional elasticity
Summary
Elasticity model in cylindrical coordinates In cylindrical coordinates, we have
ˆr + uθ e ˆθ + uz e ˆz . u = ur e ¯ is derived by Then the displacement-gradient tensor ∇u ˆr , e ˆθ as well as the differentiating the variable unit vectors e coefficients of the three unit vectors. Small-strain components in cylindrical coordinates are ur ∂uz ∂ur θ Eθθ = 1r i∂u Ezz ∂r ∂θ + r = ∂z ∂uθ uθ 1 ∂ur 1 ∂uθ 1 ∂uz + − E = + θz r ∂θ ∂r r r ∂θ 2 ∂z ∂ur 1 ∂uz Ezr = 2 ∂r + ∂z .
Errh=
Er θ =
1 2
(12)
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Introduction
Plane elasticity
Three-dimensional elasticity
Summary
Plane stress equations in polar coordinates Plane-strain components will be denoted for u = uer + veθ u eθ = 1r ∂v by er = ∂u ∂r ∂θ +r 1 1 ∂u ∂v v er θ = 2 r ∂θ + ∂r − r
(13)
Hooke’s law in polar coordinates σr τr θ σz
= λe + 2Ger = 2Ger θ = ν(σr + σθ .
σθ = λe + 2Geθ where e = er + eθ (14)
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Introduction
Plane elasticity
Three-dimensional elasticity
Summary
Lamé solution for cylindrical tube (1) Consider a tube long in the z-direction, loaded by internal pressure pi and external pressure po , with negligible body forces, and assume plane deformation with radial symmetry, independent of z and θ in the plane region a ≤ r ≤ b. The Navier equations then become, with uθ = uz = 0, ur = u, simply d 1 d (ru) = 0. (15) (λ + 2G) dr r dr Two integrations with respect to r then give u = Ar +
B . r
(16)
In polar coordinates we get er =
∂ur B =A− 2 ∂r r
eθ =
ur B =A+ 2 r r
er θ = 0
(17)
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Introduction
Plane elasticity
Three-dimensional elasticity
Summary
Lamé solution for cylindrical tube (2) Hooke’s law gives = 2Aλ + 2GA − 2GB r2 = 2Aλ + 2GA + 2GB 2 r = 0 σz = 4Aν(λ + G).
σr σθ τr θ
(18)
We apply the boundary conditions at r = a and r = b. r r
= a = b
−pi −po
= 2A(λ + G) − = 2A(λ + G) −
whence 2GB
=
2(λ + G)A =
(pi −po )a2 b2 b2 −a2 pi a2 −po b2 b2 −a2
2GB a2 2GB b2
(19)
) (20)
so that the stress distribution is σr
=
σθ
=
σz
=
2 2 pi a2 −po b2 o a b − bp2i −p b2 −a2 −a2 r 2 2 2 pi a2 −po b2 o a b + bp2i −p b2 −a2 −a2 r 2 2ν(pi a2 −po b2 ) τr θ = 0. b2 −a2
(21)
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Introduction
Plane elasticity
Three-dimensional elasticity
Summary
Outline 1
Introduction
2
Plane elasticity Rectangular coordinates Cylindrical and polar coordinates
3
Three-dimensional elasticity Direct solution of Navier’s equation Examples
4
Summary
/centre for analysis, scientific computing and applications
Introduction
Plane elasticity
Three-dimensional elasticity
Summary
Outline 1
Introduction
2
Plane elasticity Rectangular coordinates Cylindrical and polar coordinates
3
Three-dimensional elasticity Direct solution of Navier’s equation Examples
4
Summary
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Introduction
Plane elasticity
Three-dimensional elasticity
Summary
Direct solution of Navier equations
(λ + G)
∂ 2 uk ∂e + G∇2 uk + ρbk = ρ 2 ∂Xk ∂t
where e =
∂uk . ∂Xk
2
Already in 1969, for elastostatics, with ∂∂tu2k = 0, the solution of such a set of three equations in three dimensions by finite-difference or finite-element methods is beginning to be a possibility. An advantage of direct solving the 3D problem instead of the less complex 2D problem is that strains can be obtained in terms of the first partial derivatives of the displacement field.
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Introduction
Plane elasticity
Three-dimensional elasticity
Summary
The Helmholtz representation Each continuously differentiable vector field u can be represented as u = ∇φ + ∇ × ψ.
(22)
For definiteness there is also the requirement ∇ · ψ ≡ 0. Equations of motion in terms of the potentials: (λ + 2G)(∇2 φ),k + Gekrs (∇2 ψs ),r = ρ(
∂2ψ ∂2φ ),k + ρekrs ( 2 ),r . 2 ∂t ∂t
Specific solutions also satisfy the wave equations. ∇2 φ =
1 ∂2φ c12 ∂t 2
∇2 ψ k =
1 ∂ 2 ψk . c22 ∂t 2
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Introduction
Plane elasticity
Three-dimensional elasticity
Summary
Papkovich-Neuber potentials When the Helmholtz representation is substituted into the elastostatic Navier Eq., we get ∇2 [α∇φ + ∇ × ψ] = −
ρb , G
where α = 2(1−ν) 1−2ν . The Papkovic-Neuber potentials are φ0 and Φ, which is defined by Φ = α∇φ + ∇ × ψ. Then we get Poisson’s equations ( ∇2 Φ = − ρGb ∇2 φ = ρb·r 0
G
The solution can be found by Green’s formula. In general, the potentials satisfy complicated boundary conditions. /centre for analysis, scientific computing and applications
Introduction
Plane elasticity
Three-dimensional elasticity
Summary
Green’s formula (1)
Two sufficiently differentiable functions f and g in volume V bounded by S satisfy Green’s Second Identity: Z Z 2 ∂g ∂f f −g dS = f ∇ g − g∇2 f dV . (23) ∂n ∂n S V Let P be an arbitrary field point and Q a variable source point. Then Green’s formula expresses the value fP as follows: Z Z ∂ 1 1 2 1 ∂f −f dS − ∇ fdV . 4πfP = r ∂n ∂n r r 1 1 V 1 S
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Introduction
Plane elasticity
Three-dimensional elasticity
Summary
Green’s formula for the half-space In potential theory the Green’s function G(P, Q) for a region is a symmetric function of the form G(P, Q) =
1 + g(P, Q) ∇2 g = 0. r1
where r1 = kP − Qk. We obtain the formula for f in terms of Green’s function: Z Z ∂G 4πfP = − f dS − G∇2 fdV . S ∂n V Let Q2 be the image point of Q in the XY-plane and let r2 = kP − Q2 k. For the half-space, it is possible to determine G(P, Q): G(P, Q) = f (P) = −
1 ∂ 2π ∂Z
Z S
1 1 − . r1 r2
f 1 dS − r0 4π
Z
G∇2 fdV
V
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Introduction
Plane elasticity
Three-dimensional elasticity
Summary
Outline 1
Introduction
2
Plane elasticity Rectangular coordinates Cylindrical and polar coordinates
3
Three-dimensional elasticity Direct solution of Navier’s equation Examples
4
Summary
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Introduction
Plane elasticity
Three-dimensional elasticity
Summary
Normal traction problem Given on part S1 of the boundary S (z = 0) of the half-space a distributed normal pressure of intensity q, the traction boundary conditions are Tzz = −q
Tzx = Tzy = 0
on S1 .
with zero tractions on the remainder of the boundary z = 0. We seek the Papkovich-Neubich potentials, assuming that φ1 ≡ φ2 ≡ 0, such that ˆz − u = φ3 e
1 ∇(φ0 + zφ3 ) and ∇2 φ0 = ∇2 φ3 ≡ 0. 4(1 − ν)
They can be found by using Green’s formula for the half-space.
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Introduction
Plane elasticity
Three-dimensional elasticity
Summary
Boussinesq problem of concentrated normal force on boundary of half-space This problem is solved by taking limit S1 → O and q → ∞ such that R limS1 →O S1 qdS = P, which is a finite concentrated load at O in the positive z-direction.
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Introduction
Plane elasticity
Three-dimensional elasticity
Summary
Solution of Boussinesq problem
We obtain
φ2 (x, y , z) = (1−ν)P πGR (1−ν)(1−2ν)P ∂ [φ (x, y , z)] = 0 ∂Z πGR φ0 (x, y , z) = (1−ν)(1−2ν)P log(R + z) πG
Now also the displacements and tractions can be computed.
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Introduction
Plane elasticity
Three-dimensional elasticity
Summary
Outline 1
Introduction
2
Plane elasticity Rectangular coordinates Cylindrical and polar coordinates
3
Three-dimensional elasticity Direct solution of Navier’s equation Examples
4
Summary
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Introduction
Plane elasticity
Three-dimensional elasticity
Summary
Summary
Navier’s equations Plane elasticity Airy stress function Plane elasticity in polar coordinates Lamé solution for tube Helmholtz representation Papkovich-Neuber potentials Boussinesq problem
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Introduction
Plane elasticity
Three-dimensional elasticity
Summary
Literature. L.E. Malvern: Introduction to the mechanics of a continuous medium, Prentice-Hall, 1969, pp 497-568.
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Introduction
Plane elasticity
Three-dimensional elasticity
Summary
Literature. L.E. Malvern: Introduction to the mechanics of a continuous medium, Prentice-Hall, 1969, pp 497-568. Y.C. Fung: Foundations of solid mechanics, Englewood Cliffs, N.J.: Prentice-Hall, Inc., 1965. A.E. Green, W. Zerna: Theoretical elasticity, London: Oxford University Press, 1954. A.E.H. Love: Mathematical theory of elasticity, 4th ed. New York: Dover Publications, Inc., 1944. S. Timoshenko, J.N. Goodier: Theory of elasticity, 2nd ed. New York: McGraw-Hill Book Company, 1951. H.M. Westergaard: Theory of elasticity and plasticity, New York: Dover Publications, Inc., 1952. /centre for analysis, scientific computing and applications