Approximate theories for solids with special shapes:
rods, beams, membranes, plates and shells
10.5 Motion and Deformation of thin shells General theory
The figure illustrates the problem to be solved. The solid of interest is a shell with uniform thickness h. The shell’s thickness is assumed to be much smaller than any relevant in-plane dimension. The exterior surface of the shell is subjected to a prescribed distribution of traction, while the edge of the shell may either be supported so as to constrain its motion, or may be subjected to prescribed forces. Our objective is to calculate the internal forces in the shell, and to compute its deformed shape.
To specify the position of a point on the mid-plane of the undeformed shell, we introduce a convenient curvilinear coordinate system (examples include cylindrical or spherical polar coordinates). Note that need not necessarily be distances along the surface: for example, for a cylindrical shell, we would use the axial distance z and the angle as the coordinate system.
The position vector of a material particle on the mid-section of the initial shell is denoted by
To characterize the orientation of an arbitrary point in the undeformed shell, we introduce three basis vectors , with
Thus, are tangent to the coordinate lines in the undeformed shell, and is a unit vector perpendicular to the mid-section of the shell. This basis is called the covariant basis or natural basis for the coordinate system. Note that the basis vectors are not unit vectors, and are not, in general, orthogonal.
Because are not orthogonal, it is convenient to introduce a second set of basis vectors defined so that
where is the Kronecker delta symbol (the index i has been raised to match the indices on the basis vectors), i.e. for i=j and zero otherwise. This second triad of vectors is called the contravariant basis or reciprocal basis for the coordinate system. The contravariant basis vectors can be constructed by taking cross products of the covariant basis vectors, as follows
The position vector of an arbitrary point in the undeformed shell can be expressed as , where is the perpendicular distance of the material particle from the mid-section of the shell.
After deformation, the mid-section of the shell is deformed to another smooth surface. The point that lies at on the mid-section of the undeformed shell moves to a new position after deformation.
To characterize the orientation of the deformed shell, we introduce three basis vectors , with . Now, are tangent to the coordinate lines in the deformed shell, and is a unit vector perpendicular to the mid-section of the deformed shell. We can introduce a reciprocal basis in exactly the same way as for the undeformed shell.
A few special vectors and tensors, such as the angular velocity of the shell, and the internal stress couple in the shell are most conveniently expressed in terms of vectors or . Special symbols will not be introduced for these vectors; they will always be written out as a cross product.
In this section we introduce some additional notation that helps deal with the complicated sets of basis vectors that characterize the deformation of a shell.
Vectors can be expressed as linear combinations of some subset of the twelve possible basis vectors , , or . For example, we can write an arbitrary vector a as
Here, the coefficients are called the contravariant components of a in , and are called the covariant components of a. Note that the contravariant components are coefficients of the covariant basis vectors, and vice-versa.. The reason for this confusing terminology is given below. Note also that the components do not in general have the same units as the vector, because the basis vectors may have length dimensions.
The various components of a can be expressed as
To see the first result, take dot products of with and recall that . The `contravariant’ and `covariant’ terms assigned to and refer to the fact that they represent projections of the vector a onto the contravariant and covariant basis vectors, respectively. The raised and lowered indices for vector components follow the same convention: raised indices indicate contravariant components, while lowered indices represent covariant components.
Tensors can also be expressed as sums of nine dyadic products of various combinations of basis vectors. For example, if S is a tensor we could write
or equivalent expressions in terms of and/or . Here are called contravariant and covariant components of S, respectively, while are called mixed tensor components. The dot that appears before the indices in the mixed tensors is introduced to identify whether the index should be associated with the first or second basis vector in the dyadic product (the dot appears before the index associated with the second basis vector). For symmetric tensors, the dot can be dropped.
The various components of S can be regarded as projections of the tensor onto the contravariant or covariant basis vectors, as , with similar results for the mixed components.
Once again, it is important to note that the components do not have a convenient physical interpretation. In general the components do not even have the same units as the tensor itself, because the basis vectors themselves have units.
The various sets of components can be related by defining the components of the fundamental tensor or metric tensor g as follows
We can define in terms of and/or in the same way. With these definitions, we see that the covariant and contravariant basis vectors are related by
The metric tensor can be expressed as the sum of three dyadic products , with a similar expression for . From this expression we see that
g is therefore its own inverse it is a representation of the identity tensor.
An infinitesimal line element that lies in the mid-section of the undeformed plate can be expressed in terms of infinitesimal changes in the coordinates as
The length ds of can be computed as
This expression is known as the first fundamental form for the surface. A similar expression can be constructed for the deformed surface.
The variation of the normal vectors and with position in the mid-plane of the shell play a particularly important role in describing the shape and deformation of the shell, because they characterize its curvature and bending. To quantify this variation, let and be the vectors normal to the surface at positions and in the undeformed shell, with a similar notation for the deformed shell. We introduce symmetric curvature tensors and that satisfy
The curvatures and are called surface tensors, because they transform like tensors under changes of surface coordinates.
The curvature components can be expressed in terms of their covariant, contravariant or mixed components. This can be used to deduce expressions such as
The curvature components can be calculated from the position vector of the mid-plane of the shell, using any of the following expressions
where . The mixed components follow as , where are the components of the metric tensor defined in Sect 10.5.2.
The magnitude of the curvature of a shell is quantified by principal curvatures - these are simply the principal values of . The mean curvature , and Gaussian curvature are also used.
We will also need to calculate the variation of the remaining basis vectors with position in the surface. These are quantified by Christoffel symbols of the second kind which satisfy
The Christoffel symbols are functions of position on the surface, and can be related to the position vector of the mid-plane of the shell and its curvature components as
Some relationships between the time derivatives of these various kinematic quantities are also needed in subsequent calculations. The rate of change in shape of the shell can be characterized by the velocity of its middle surface . The velocity vector can be described as components in any of the various bases: the representation is particularly useful.
The time derivatives of the basis vectors are a convenient way to characterize the rate of change of bending of the shell. These are related to the velocity of the shell’s mid-plane by
We will also need to calculate the time derivative of the vector normal to the mid-plane of the shell . Since is a unit vector, its time derivative can be quantified by an angular velocity vector , defined so that
The components of can be related to as
Recalling that , we see also that
Finally we may write this as
The components can also be regarded as the components of the angular velocity vector in a basis in the sense that
The time derivative of the curvature tensor , is related to by
The components of the time derivative of curvature can therefore be expressed in terms of as
It is important to note that are not equal to the time derivatives of the curvature components.
We will also need to characterize the linear and angular acceleration of the shell. The linear acceleration can be quantified by the acceleration of the mid-plane .
The angular acceleration of the shell can be characterized by the angular acceleration of the normal to its mid-plane, . The angular acceleration can be related to the acceleration of the mid-plane of the shell as follows
where we have used and noted that to obtain the final result.
It is convenient to express in the form , where and the can be related to the velocity and acceleration of the mid-plane of the shell as follows
These results show that
where we have used and noted to obtain the second line.
HEALTH WARNING: The sign convention used to characterize the curvature of a shell can be confusing. In the convention used here, a convex surface has positive curvature. For example, a spherical shell with coordinate system chosen so that points radially out of the sphere would have two equal positive principal curvatures. The mathematical analysis of curved surfaces usually uses the opposite sign convention for curvature, and a few texts on shell theory use curvature measures with the opposite sign to the one used here.
The position vector of a material point in the shell before deformation can be expressed as , where is the distance of the material particle from the mid-section of the shell.
After deformation of the material point that has coordinates in the undeformed shell moves to a new position, which can be expressed as
where . This is a completely general expression. We now introduce a series of approximations that are based on the assumptions that
With this in mind, we assume that can be approximated by a function of the form
where can be regarded as the first term in a
Several versions of plate theory exist, which use different approximations for the shear deformation. Here, we will present only simplest version, known as Kirchhoff shell theory, which is to assume that . This implies that material fibers that are perpendicular to the mid-plane of the shell remain perpendicular to the mid-plane of the deformed shell, and the shell does not change its thickness. This reduces the displacement field to
The velocity field can be approximated as
while the acceleration is
where and denote the angular acceleration and angular velocity of the unit vector normal to the mid-plane of the plate.
HEALTH WARNING: In addition to using this approximation to the displacement and velocity field, Kirchhoff shell theory assumes that the transverse stress vanishes in the shell. Strictly speaking, this is inconsistent with the deformation. A more rigorous approach would be to introduce a uniform transverse strain , which could be calculated as part of the solution. However, this approach yields results that are essentially indistinguishable from the plane-stress approximation.
10.5.5 Approximating the deformation gradient
The deformation gradient can be approximated as
where and are the metric tensors for shell before and after deformation, and and are the curvature tensors for the mid-surface of the shell before and after deformation, respectively. The three terms in the second formula for F can be interpreted as (i) the effects of in-plane stretching of the shell; (ii) the effects of bending; and (iii) the effects of a change in the shell’s thickness.
Derivation By definition, the deformation gradient relates infinitesimal line elements in the shell before and after deformation by . We wish to construct a tensor with these properties.
where we have used
This can be re-written as
We can substitute this result into (2) above to see that
and the deformation gradient can be read off as the coefficient of dx.
To see this, multiply out , recall that and neglect the term of order . Finally, substitute this approximation into the formula for F, multiply out the terms and neglect terms to obtain the approximation for F.
It is straightforward to calculate any other deformation of interest from the deformation gradient. A few examples that will be used in calculations to follow are listed below.
The inverse of the deformation gradient can be approximated by
The velocity gradient tensor , which relates the relative velocity of two material particles at positions and in the deformed shell as can be approximated by
where are the covariant components of the time derivative of the surface curvature tensor.
The Lagrange strain tensor can be approximated by
where terms of order have been neglected, we have used and .
When we write constitutive equations relating forces to deformations, it is convenient to introduce two new strain measures defined as follows:
where and . The tensor quantifies length changes of infinitesimal material elements in the mid-plane of the shell, in the sense that the lengths and of a line element before and after deformation are related by
The figure shows a generic cross-section of the shell, in the deformed configuration. To define measures of internal and external forces acting on the shell, we define the following variables
A basis with vectors chosen following the scheme described in 10.5.1. Vector and tensor quantities will be quantified by their contravariant components in this basis
The body force acting on the rod , or in component form
The tractions acting on the exterior surface of the shell . It is convenient to define separate variables to characterize the tractions acting on the various parts of the shell, as indicated in the sketch: the upper surface of the shell (denoted by ) is subjected to traction ; the lower surface is subjected to , while the surface around the edge of the shell is subjected to
The Cauchy stress within the shell .
External forces and moments acting on the shell are characterized by
The vector q is work conjugate to the angular velocity of the normal to the mid-plane of the shell, in the sense that is the rate of work done by the external couple per unit area of the shell’s mid-plane.
Internal forces and moments within the shell are characterized by three surface tensors , and , To visualize their physical significance, suppose that the shell is cut through so as to expose an internal surface, which lies perpendicular to the mid-plane of the shell. Let denote a unit vector normal to the internal surface, and let ds denote an infinitesimal line element that lies in the both the exposed surface and the mid-plane of the shell. The exposed surface is subjected to a distribution of traction, so that an small element of area with dimensions is subjected to a resultant force and resultant moment . These forces and moments are related to , and as outlined below:
1. The in-plane stress resultant tensor quantifies internal forces that tend to stretch and shear the shell in its own plane. It is related to the internal tractions by , and its components can be calculated from the stress distribution in the shell as
2. The transverse shearing stress tensor quantifies internal forces that act to impose the constraint that material fibers that are perpendicular to the mid-plane of the shell before deformation remain perpendicular to the mid-plane after deformation. Strictly speaking, in Kirchoff shell theory it is a Lagrange multiplier, but can be regarded as quantifying the transverse shear force . Its value cannot be computed from the deformation of the shell, because the transverse shearing has been neglected: instead, it must be determined by solving the equilibrium equations given in the next section.
3. The internal moment tensor characterizes internal bending and twisting moments in the shell. It is related to the moment acting on internal through-thickness sections of the shell by . The components can be calculated from the internal stresses in the shell as
The tensor is work conjugate to the gradient of the angular velocity of the normal to the mid-plane of the shell , or alternatively, to the rate of change of curvature in the sense that
is the rate of work done by M per unit area of the mid-plane of the shell.
We consider a shell with thickness h and mass density . The internal forces and moments must satisfy
The operator represents the surface divergence, T, V and M are the internal forces defined in Sect 10.5.7; p and q are the external force and couple per unit area acting on the shell, a is the acceleration of the mid-plane and is the angular acceleration of the unit vector normal to the mid-plane of the shell. The two equations can be interpreted as linear and angular momentum balance for an infinitesimal element of the shell. Note that:
1. If the system is in static equilibrium, the right hand sides of all the equations of motion are zero.
2. In addition, in many dynamic problems, the right hand sides of the angular momentum balance equations may be taken to be approximately zero. For example, the rotational inertia may be ignored when modeling the vibration of a shell. The rotational inertia terms can be important if the shell is rotating rapidly: for example, they would influence the out-of-plane vibration of a spinning disk.
The equations of motion can also be expressed in terms of components of the various force and moment tensors by substituting , , , , and and recalling that
The result is
The last equation shows that the stress resultant and moment tensors are not symmetric. The asymmetry is small, and is ignored in simplified versions of shell theory. However, there are a few special shell geometries (a cylindrical shell subjected to torsional loading is one example) where neglecting the asymmetry can lead to substantial errors.
Edge boundary conditions for a shell are complicated and confusing. To understand them, it is helpful to visualize the possible types of motion that can occur at the edge of a shell. The edge of the shell is characterized by a curve C that lies in the mid-plane of the shell, encircling in a counterclockwise sense. We let denote arc-length measured around C from some convenient origin, and use and denote unit vectors tangent and normal to C. Elements of the shell that lie on C have four independent degrees of freedom, as follows:
The motion of the edge of the shell can be prescribed by constraining one or more of these degrees of freedom. Alternatively, the edge of the shell can be subjected to one or more of four generalized forces, which are work-conjugate to these degrees of freedom. The generalized forces can be expressed in terms of the forces and couples acting on the edge of the shell as
The four boundary conditions can be expressed in terms of these forces as follows:
Derivation: Measures of internal force, the equilibrium equations, and the boundary conditions emerge naturally from the principle of virtual work. The principle of virtual work states that, for any deformable solid that is in static equilibrium, the Cauchy stress distribution must satisfy
for all virtual velocity fields and compatible velocity gradients . The virtual velocity field in the shell must have the same general form as the actual velocity, as outlined in Section 10.2.4. The virtual velocity can therefore be characterized by the virtual velocity of the mid-plane of the shell . It is convenient to introduce the time derivative of the normal vector to the plate’s mid-plane as an additional kinematic variable, which must of course be compatible with . We will show the following:
The virtual work principle can be expressed in terms of the generalized deformation measures and forces defined in the preceding sections as
If the virtual work equation is satisfied for all and compatible , then the internal forces and moments must satisfy the following equilibrium equations
as well as the boundary conditions listed above.
The last equilibrium equation does not emerge from the virtual work principle. Instead, this equation is a consequence of the symmety of the Cauchy stress tensor , as shown below. It is automatically satisfied if the components and are calculated by integrating the stresses through the thickness of the shell. However, for some statically determinate shell problems it is possible to avoid evaluating these integrals explicitly, in which case the equilibrium equation is useful.
Expressing the virtual work equation in terms of generalized force measures is a straightforward but lengthy algebraic exercise.
Recall that , so that
Therefore , where the term of order has been neglected. Substituting this result into the volume integral yields
where the area integral is understood to be taken over the mid-plane of the shell.
Following the procedure in step (1), the integrals over and can be expressed in terms of integrals taken over the mid-plane of the shell as
The integral over can be reduced to a line integral taken around the curve(s) bounding the edge of the shell, as
The procedure to derive this result is very similar to the steps required to simplify the volume integral and is left as an exercise.
To reduce this to a scalar combination of the components of the various tensors and vectors, substitute , together with the kinematic formulas:
with the result
Substituting the equations in (4) and (5) into the virtual work equation gives the first result.
Next, we show that the equilibrium equations and boundary conditions follow from the virtual work principle.
for all admissible variations , and . The second integral can simply be added to the virtual work equation to ensure compatibility of and
where is the outward normal to C. To see how to use this theorem, consider
The integrand can be re-written as
Applying the surface divergence theorem to the first term on the right hand side of this equation shows that
Finally, substituting and and remembering to differentiate the basis vectors gives the component form
This equation must be satisfied for all and , which immediately gives the equilibrium equations.
Here, and represent unit vectors tangent and normal to C, and is an independent degree of freedom that represents the rotation of the shell about . Finally, we integrate by parts to see that
(The terms associated with the ends of C vanish because C is a closed curve).
This condition must be satisfied for all , which gives the boundary conditions.
Finally, we must derive the last equilibrium equation . Using the definitions of , and noting that , it is straightforward to show that
Adding these two equations gives the last equilibrium equation.
The internal forces in a shell are related to its deformation by the stress-strain law for the material. Here, we give force-deformation equations for an isotropic elastic shell which experiences small shape changes (but possibly large rotations).
Shape changes are characterized using the following deformation measures, defined in Sections 10.5.6
Internal forces are characterized using the stress resultant tensor components and internal moment components defined in Section 10.5.7.
The shell is assumed to have a uniform thickness h, and is assumed to be made from an isotropic, linear elastic solid, with Young’s modulus E and Poisson’s ratio . We assume for simplicity that the shell is homogeneous, and neglect thermal expansion (the effects of thermal expansion are included an example problem solved in Section 10.7.4)
It is convenient to introduce a plane stress elasticity tensor with components
The force-deformation relations can then be expressed as
For all but a few very rare shell geometries these expressions may be approximated by
where is the in-plane component of the metric tensor associated with the undeformed shell (this replaces the identity tensor in the Cartesian version of the constitutive equations)
Substituting the formula for stress components into this expression and integrating through the thickness of the shell gives
Here, terms of order have been neglected.
Substituting the formula for stress components into this expression and integrating through the thickness of the shell gives
where terms of order and higher have been neglected.
10.5.10 Strain energy and kinetic energy of an elastic shell
It is useful to express the strain energy and kinetic energy of a deformed shell in terms of the motion and deformation of its mid-plane. To this end:
The total strain energy of the shell can be calculated as
For all but a very few special shell geometries this result may be approximated by
The kinetic energy can be calculated using the formula
The second term in the expression for the kinetic energy represents the rotational energy. In many practical problems, such as vibration of a shell, the rotational energy can be neglected.
Derivation: The strain energy density in the shell is given by , where is the material stress tensor and E is the Lagrange strain tensor, defined in the preceding section. The stress can be expressed in terms of the strain using the constitutive equation, while the strain can be expressed in terms of and using step (3) in Section 10.5.8. Integrating over the volume of material in the shell, and evaluating the integral through the shell’s thickness explicitly gives the result stated. The kinetic energy is calculated using the formula for the velocity field in 10.2.8.
(c) A.F. Bower, 2008