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Deformable Bodies
Deformation
• Given a rest shape x and its deformed configuration p(x), how large is the internal restoring force f(p)?
• To answer this question, we need a way to measure deformation
x p(x)
• Measurement of deformation
• Measurement of elastic force
• Constitutive law
• Finite element method
Displacement field
• Displacement field directly measures the difference between the rest shape and the deformed shape
• It’s not rigid-motion invariant. For example, a pure translation p = x + 1 results in nonzero displacement field u = 1
Displacement gradient
• Displacement gradient is a matrix field
• Need to compute deformation gradient
• Both displacement gradient and deformation gradient are translation invariant but rotation variant
Green’s strain• Green’s strain can be defined as
• Green’s strain is rigid-motion invariant (both translation and rotation invariant)
rpTrp� I = (RS)T RS� I = ST RT RS� I = ST S� I
Cauchy’s strain
• When the deformation is small, Cauchy’s strain is a good approximation of Green’s strain
• Is Cauchy’s strain rigid motion invariant?• Consider a point at rest shape x = (x, y, z)T and its deformed
shape p = (-y, x, z)T, what is the Cauchy’s strain for this deformation?
• Measurement of deformation
• Measurement of elastic force
• Constitutive law
• Finite element method
• Strain measures deformation, but how do we measure elastic force due to a deformation?
• Stress measures force per area acting on an arbitrary imaginary plane passing through an internal point of a deformable body
• Like strain, there are many formula to measure stress, such as Cauchy’s stress, first Piola-Kirchhoff stress, second Piola-Kirchhoff stress, etc
Elastic force
Stress
• Stress is represented as a 3 by 3 matrix, which relation to force can be expressed as
• da is the infinitesimal area of the imaginary plane upon which the stress acts on
• n is the outward normal of the imaginary plane.
Cauchy’s stress
• All quantities (i.e. f , da and n) are defined in deformed configuration
• Consider this example, what is the force per area at the rightmost plane?
Cauchy’s stress
• The internal force per area at the right most plane is
• σ11 measures force normal to the plane (normal stress)• σ21 and σ31 measure force parallel to the plane (shear stress)
• Measurement of deformation
• Measurement of elastic force
• Constitutive law
• Finite element method
• Constitutive law is the formula that gives the mathematical relationship between stress and strain
• In 1D, we have Hooke’s law
• Constitutive law is analogous to Hooke’s law in 3D, but it is not as simple as it looks
Constitutive law
Constitutive law
• What is the dimension of C?
2
4"11 "12 "13"21 "22 "23"31 "32 "33
3
5
Materials• For a homogeneous isotropic elastic material, two independent
parameters are enough to characterize the relationship between stress and strain
• E is the Young’s modulus, which characterize how stiff the material is
• ν is the Poisson ratio, ranging from 0 to 0.5, which describe whether material preserves its volume under deformation
• Measurement of deformation
• Measurement of elastic force
• Constitutive law
• Finite element method
Finite element method
• So far we view deformable body as a continuum, but in practice we discretize it into a finite number of elements
• The elements have finite size and cover the entire domain without overlaps
• Within each element, the vector field is described by an analytical formula that depends on positions of vertices belonging to the element
Tetrahedron
• Rest shape of a tetrahedron is represented by x0, x1, x2, x3• Deformed shape is represented by p0, p1, p2, p3• Any point x inside the tetrahedron in the rest shape can be
expressed using the barycentric coordinate
Barycentric coordinates
• FEM assumes that deformed shape is linearly related to rest shape within each tetrahedron
• Therefore, p(x) can be interpolated using the same barycentric coordinates of x
• p(x) can also be computed as
Elastic force
• To simulate each vertex on a tetrahedra mesh, we need to compute elastic force applied to vertex
• Based on p(x), compute current strain of each tetrahedron• Use constitutive law to compute stress• For each face of tetrahedron, calculate internal force:
• A is the area of the face and n is the outward face normal• Distribute the force on each face to its vertices
Linear FEM
• Assuming deformation is small around rest shape, it is valid to use Cauchy’s strain and calculate face normal and area using rest shape
• Simplified relationship between internal force and deformation
• K can be pre-computed and maintain constant over time
Corotational FEM
• When object undergoes rotation, the assumption of small deformation is invalid because Cauchy’s strain is not rotation invariant
• Corotational FEM is an effective method to eliminate the artifact due to rotation
• first extract rotation R from the transformation• rotate the deformed tetrahedron to the unrotated frame RTp• calculate the internal force K(RTp − x)• rotate it back to the deformed frame: f = RK(RTp − x)
Corotational FEM