Kinematics: Configurations and Motion of a Body

• Kinematics studies motion without regard to the motion-causing forces.
• A solid understanding of kinematics is essential for theorizing how a body responds to forces, moments, or thermal loads.
• Describing a deformable body's kinematics can be simplified by regarding it as an infinite set of particles.

Configurations and Motion of a Body

• A body can be conceptualized as an unlimited set of particles.
• As a body moves the particles remain fixed, only their positions change with time.
• The set of all particle positions relative to a fixed frame at an instant is a "configuration" of the body.
• The "reference configuration" is the body's configuration at a reference time.
• The "current configuration" is the body's configuration at a future time t.
• Motion of the body can be defined using the function Φ, to map points in the reference configuration to their corresponding points in the current configuration: x = Φ(X, t)
  • X is the particle's position in the reference configuration.
  • x is the particle's position in the current configuration at time t.
• The coordinate frame for reference configuration particle positions, are called "material coordinates".
• The coordinate frame for current configuration particle positions are called "spatial coordinates".
• In solid mechanics, the reference configuration may be called "undeformed configuration", and the current configuration the "deformed configuration."
• Static deformation simplifies motion as a function of material coordinates only: x = Φ(X)

Displacement, Velocity, and Acceleration Fields

• Displacement is the change in position of a particle relative to the reference configuration.
• Assuming spatial and material coordinate frames share the same origin, displacement is: u(X, t) = Φ(X, t) – X
• Velocity of a particle is defined as the time rate of change of position: v(X, t) = ∂Φ/∂t (X, t)
• The velocity of the body is now a vector field.
• The acceleration field is the second time derivative of the motion: a(X, t) = ∂²Φ/∂t² (X, t)

Special Case of Rigid Body Motion

• Rigid body motion is a motion where the distance between any two points in the body are invariant with respect to time.
• Given points Xa and Xb in the reference configuration, there is a constant c such that ||Φ(Xa, t) – Φ(Xb, t)|| = c for all time t.
• The motion of a rigid body can be represented as a rotation about Xo followed by a translation of Xo: Φ(X, t) = do(t) + R(t) · (X – Xo)
  • do is the displacement of Xo.
  • R is the rotation tensor (rank-2) about Xo.
• Any rigid body motion can be written as a rotation about the origin by a translation: Φ(X,t) = d(t) + R(t) · X
• Finite rotations can be represented as a rotation about a single axis by some angle θ.
• An expression is found for R in terms of n̂ and θ: x = (cos θ)X + (1 − cos θ) (X · n̂)n̂ + (sin θ)n̂ × X
• The first term can be rewritten as: (cos θ)X = (cos θ)I · X
• The second term rewritten: (1 − cos θ) (X· n̂) n = (1 − cos θ)n (n · X) = (1 − cos θ)n ⊗ n • X
• The third term of may seem less straightforward due being acted upon by a cross product.
• N can be defined by the equation: N = εijkêi ⊗ êk
• The third term can be rewritten as: n x X = N · X
• Comparing, the value if R reveals: R = (cos θ)I + (1 − cos θ)n ⊗ n + (sin θ)N

Stretching of Material Curves

• A "material curve" is a set of material points forming a curve within a body, deforming with the body.
• A curve in space can be represented as a vector-valued function r of a single parameter s.
• A tangent to this curve is given by dr/ds.
• If s measures the distance along the curve, then dr/ds is a unit vector.
• A material curve is defined in the undeformed configuration by a vector valued function r of a parameter s.
• The value of r(s) is the material position of a particle that lies on the curve at distance s along the curve.
• The corresponding spatial position of a particle at s on the curve is Φ(r(s), t).
• Distance along the material curve changes as the body deforms
  • Let l be the distance along the curve in the deformed configuration.
• The "stretch ratio" λ quantifies stretching, given by λ = dl/ds.
• The stretch ratio will be a function of s, and the amount the length has changed.
• Let z be path function tracing the material curve in the deformed configuration as a function of l at some time t, z(l(s)) = Φ(r(s),t)
• Isolate λ, and the gradient of appears frequently in solid mechanics.
• The "deformation gradient" F: F = ∂Φ/∂X
• The equation λ² = (dr/ds)ᵀ · (∂Φ/∂X)ᵀ· (∂Φ/∂X) · (dr/ds) may now be written λ² = (dr/ds)ᵀ· Fᵀ · F · (dr/ds) -The quantity FT. F appears frequently and is the "right Green-Cauchy deformation tensor" C as C = FT· F
• Equation can be written as λ² = (dr/ds)ᵀ· C · (dr/ds)
• A only depends on dr/ds, a unit tangent vector, and not r itself - let g = dr/ds
• Stretch ratio can then concisely be expressed as λ = √(g·C·g)

Angle Between Material Curves

• Imagine drawing two orthogonal lines through the undeformed configuration of a body. How does the angle between these two lines change as the body undergoes motion ?
• Pick orthogonal directions on₁ and on₂, and two material curves:
  • U(s₁) = X₀ + s₁n₁
  • V(s₂) = X₀ + s₂n₂
• In the deformed configuration, the material curves are given by:
  • u(l₁(s₁)) = Φ(U(s₁), t)
  • v(l₂(s₂)) = Φ(V(s₂), t)
• In general, the material curves will not be straight lines in the deformed configuration.
• The angle θ between the curves can be computed using the dot product: cos θ =(du/dl₁) · (dv/dl₂)
• Isolate the tangent vector du/dl₁ - where λ₁ = dl₁/ds₁ and F = ∂Φ/∂X : du/dl₁ = (1/λ₁) * F ⋅ n₁
• Simplify to: cos θ = (1/λ₁λ₂) * n₁ C n₂

Volumetric Change of Material Elements

• A "material element" is a set of particles that occupy an infinitesimal region of space within the body.
• When a body undergoes motion, the volume occupied by a material element may change, where: dV dV denote the volume of space occupied by a material element in the deformed configuration and dVo dVo the volume of space occupied by the same material element in the undeformed configuration.
• dV/dVo - relates to the motion.
• Define three material curves passing through any point X₀ aligned with coordinate lines X₁, X₂, and X₃ in the undeformed configuration:
  • U(X₁) = X₀ + X₁e₁
  • V(X₂) = X₀ + X₂e₂
  • W(X₃) = X₀ + X₃e₃
• In the undeformed configuration, a material element at X₀ occupies the differential volume dV₀ = dX₁dX₂dX₃
• In the deformed configuration, the material curves associated:
  • u(X₁) = Φ(U(X₁), t)
  • v(X₂) = Φ(V(X₂), t)
  • w(X₃) = Φ(W(X₃), t)
• The volume occupied by the material element in the deformed configuration: dV = (du × dv) · dw
• Apply the chain rule, and the du equations: du = F e₁dX₁ Likewise: dv = F e₂dX₂ and dw = F e₃dX₃
• Substitute for dV equations, and the volumetric change: dV/dV₀ = det (F)
• Recall, the change in volume of a material element may also be written in terms of the right Cauchy-Green deformation tensor C = FT· F
• dV/dV₀ = ✓ det(C)

Lagrange Strain

• The right Green-Cauchy deformation tensor C encodes information about the deformation of a body. Information includes:
• How much material lines stretch
• How much of material element change volume
• How the angle between material lines changes
• The right Green-Cauchy deformation tensor is also insensitive to rigid body motions.
• The configuration can be defined as: Φ(X, t) = d(t) + R(t) · X
• Deformation Gradient: F = R - this means C is: C = I
• Fixed by defining a rank-2 tensor by the expression C - I, leading to the Lagrange strain tensor.
• The "Lagrange strain tensor" is defined by: L = ½ (C-I)
• L₁₁ can be interpreted as a measure of the elongation of a material element in the X₁ direction. - where λ₁ is the stretch ratio of a material curve aligned with the X₁ axis in the undeformed configuration - and L₁₁ = ½ (λ₁²-1)
• The strain components L₁₁, L₂₂, L₃₃ are called "normal strains".
• Lij is a measure of the angle between two sides of a material element, if i ≠ j and Lᵢⱼ = ½ λᵢλⱼ cos θ
• It is also possible to express Lagrange strain in terms of the displacement field:
  • Displacement is defined by: u(X, t) = Φ(X, t) – X
  • Which can be expressed as: Φ(X, t) = u(X, t) + X

Infinitesimal Strain

• In deformation, the Lagrange strain may be derived.
• If the deformation is small, the Lagrange strain = approximately: L ≈ ½ ((∂u/∂X) + (∂u/∂X)ᵀ)
• Define the "infinitesimal strain" as: E = ½ ((∂u/∂X) + (∂u/∂X)ᵀ)
• Infinitesimal strain is also sometimes called "engineering strain".
• E will likely break down if a structural component has a high degree of flexibility.
• Suppose that a body undergoes a pure rigid body motion: Φ(X, t) = d(t) + R(t) · X
• The infinitesimal strain reports non-zero strain where there is no strain, as: E = ½ [R+ Rᵀ – 2I]
• Don't use E as a strain measure if finite rotation material elements is expected to occur.

Principle Strains

• Given a strain tensor L, are there directions where only direct strain occurs?
  • Is there a n where: L. n = Lₙn L
  • This can also be written as: (L - LₙI) · n = 0 - where Lₙ is an Eigenvalue and n is an Eigenvector.
• The Eigenvectors n₁, n₂, n₃ are mutually orthogonal.
• The n directions correspond to maximum direct strain and another minimum direct strain.
• The deformation may be understood as merely compression and/or extension along the principle axes n₁, n₂, n₃.