6 Linear elasticity
Characteristic for continuous solids is that a solid can possess a non-trivial stress distribution at rest. The question often is, how such an arbitrary stress distribution translates into deformation and displacement of the material. Elasticity theory for continuum materials provides an explanation for a particular class of solids. It is a very old subject. The work of Hooke dates back to 1678. Elasticity theory has been developed in parallel to fluid mechanics, however, shares many analogies in the way the continuous material is described.
Most of the journey from Hooke’s one-dimensional spring law to the three-dimensional tensor relation we will derive in §6.7 took place across the nineteenth century. Hooke published his law in 1678 in cipher: ut tensio sic vis, “as the extension, so the force.” It then took a sequence of competing programmes — Navier’s molecular theory (1820), Cauchy’s continuum stress tensor (1822, 1828), Poisson’s molecular memoir (1827), Green’s strain-energy argument (1839), and the Lamé–Clapeyron school of elasticity coefficients — before the modern continuum form of Hooke’s law settled into the version of this chapter. A well-organised survey is (Maugin 2016 ch. 1).
One persistent thread of that debate concerns the number of independent elastic constants for an isotropic material. Cauchy’s molecular theory predicted one (the Cauchy relation \(\lambda = \mu\)); Green’s energy argument and experiment forced two. The structural derivation in §6.7 produces exactly two — for a reason the molecular pioneers did not yet have access to: the representation theorem for isotropic 4-tensors.
The presentation in §§6.1–6.10 follows in large parts (Gonzalez and Stuart 2001) and (Howell, Kozyreff, and Ockendon 2009 ch. 4). The dimensional reduction in §6.11 follows (Howell, Kozyreff, and Ockendon 2009 ch. 6) and uses the asymptotic methodology of (Holmes 2013). The viscoelastic extension in §6.12 follows (Lakes 2009).
Two distinct small parameters appear in this chapter. They serve different purposes and must not be confused:
- Strain tensor \(D\) — symmetric part of the displacement gradient \(\nabla \mathbf w\) (introduced in §6.6).
- Kinematic small parameter \(\epsilon\) — magnitude of the displacement gradient, \(\partial w_i / \partial \zeta_j = \mathcal O(\epsilon)\) (introduced in §6.6). Linearises the constitutive law.
- Geometric small parameter \(\eta = h / L\) — thickness-to-span ratio of a thin structure (introduced in §6.11). Drives a second, geometric reduction from 3D to plate or beam theory.
- Plate bending stiffness \(K_b = E h^3 / [12 (1 - \nu^2)]\) (introduced in §6.11). Replaces the symbol \(D\) that the plate-theory literature conventionally uses, since \(D\) is reserved here for the strain tensor.
6.1 Hooke’s law
Assume, we have a rod of length \(l\), radius \(a\), and cross-sectional area \(A\) that is subject to a load \(P\) in longitudinal direction, see also figure
As a consequence the rod is strained, which implies a longitudinal stretching of \(\delta l\) and a cross-sectional thinning of \(\delta A\). Classical Hooke’s law states that the stretching can be determined from
\[ \underbrace{\frac{\delta l}{l}}_{\text{strain } \epsilon } = \underbrace{\frac{1}{E}}_{\substack{\text{proportionality} \\ \text{constant}}} \; \underbrace{\frac{P}{A}}_{\text{stress } \sigma }. \]
Hence, internal stress is related to longitudinal (axial) strain within the rod:
\[ \sigma = E \epsilon. \]
Hooke’s law hence states that stress and strain are proportional with a proportionality constant given by Young’s modulus \(E\). The latter characterizes a mechanical stiffness of a material. Accordingly, we have
\[ \frac{\delta a}{a} = - \nu \frac{1}{E}\,\frac{P}{A}, \]
meaning that the transverse strain \(\tfrac{\delta a}{a}\) in direction perpendicular to the loading differs from the longitudinal strain \(\tfrac{\delta l}{l}\) by a factor denoted as the Poisson ratio \(\nu\).
So far, we looked at one particular rod and one specific loading. Hooke’s law now states that the two material parameters (Young modulus \(E\) and Poisson’s ratio \(\nu\)) are - within a certain scaling and loading regime referred to as the linear elastic regime - is independent of the rod’s dimension \(l\) and \(A\) as well as of the magnitude of the loading \(P\).
In this lecture we will focus on situations, in which \(E\) is very large, e.g. \(\mathcal O ( \text{GPa})\)), such that longitudinal stretching is very small in comparison to the original length of the rod (\(\delta l \ll l\)). This means that the strain is very small: \(\tfrac{\delta l }{l} \ll 1\). For instance, if we load a steel rod of 1m length and 1cm in diameter with a mass of 1t in longitudinal direction, we will observe a longitudinal stretching in the order of \(\mathcal O(10^{-3})\) and a cross-sectional thinning in the order of \(\mathcal O(10^{-6})\). The steel rod retains its original configuration, once the load is released.
Such a material is referred to as a linear, elastic solid. The question now is: How do we carry over Hooke’s law to a continuous material body? Essentially, we are interested in the displacement of material particles in response to arbitrary, potentially time dependent loading. Can we re-write formerly derived balance laws, e.g. momentum balance \(\rho \tfrac{d}{dt} \mathbf v = \nabla \cdot \sigma\) to allow for the modeling of displacements within a solid material? How can we formulate a generalized version of Hooke’s law as a closure for the momentum balance’s right hand side \(\nabla \cdot \sigma\)? Answering these questions constitutes the roadmap for this section on linear elastic solids.
6.2 Deformation kinematics
Let’s assume, we have a material that is subject to a loading and hence deformed. Deformation is given by a mapping \[\phi : B \rightarrow B',\] also referred to as the deformation function which maps a materials original (reference) configuration \(\mathbf \zeta \in B\) is mapped onto a current (deformed) material configuration \(\mathbf x = \phi(\mathbf \zeta,t) \in B'\). Points in \(B\) are denoted by \(\mathbf \zeta = (\zeta_1,\zeta_2,\zeta_3)^T\) and refer to material coordinates, whereas points in \(B'\) are denoted by \(\mathbf x=(x_1,x_2,x_3)^T = (\phi_1(\mathbf \zeta,t),\phi_2(\mathbf \zeta,t),\phi_3(\mathbf \zeta,t))^T\) refer to deformed or spatial coordinates.
Deformation mapping \(\phi\) codes local changes in shape, for instance local stretching, which is referred to as strain. Consider an open ball \(\Omega \subset B\) of radius \(\epsilon\) centered around \(\mathbf \zeta_0\). \(\phi(\Omega)\) then refers to the ball after deformation
\[\phi(\Omega) = \{ \mathbf x \in B' | \mathbf x = \phi(\mathbf \zeta,t), \mathbf \zeta \in B \}.\]
Any relative difference of \(\Omega\) versus \(\phi(\Omega)\) as \(\epsilon \rightarrow 0\) is called strain at \(\mathbf \zeta_0\). Note, however, that \(B'\) not necessarily indicates a change in shape. It can likewise code a rotation or a translation of reference configuration \(B\).
The type of deformation can be further characterized by the deformation gradient \(F\) defined as
\[ F := \nabla \phi(\mathbf \zeta) = \frac{\partial \phi}{\partial \mathbf \zeta} \qquad \Leftrightarrow \qquad F_{ij}:=\frac{\partial \phi_i}{\partial \zeta_j}. \]
It is hence given as the derivative of each component of the deformed configuration \(\mathbf x\) with respect to each component of the reference position vector \(\mathbf \zeta\). \(F\) provides information on the local behaviour of the deformation mapping. By means of a Taylor expansion, we get
\[\triangle \mathbf x = \mathbf x - \mathbf x_0 = F(\mathbf \zeta_0) \triangle \mathbf \zeta + \mathcal O(|\triangle \mathbf \zeta|^2).\]
We note the following classics:
Pure rigid body displacement yields \(F=I\).
Example: \(x_1 = \zeta_1 + 5, x_2 = \zeta_2, x_3 = \zeta_3.\)
Pure rigid body rotation yields a deformation gradient corresponding to a rotation matrix.
Example: \(x_1 = \cos \theta \zeta_1 - \sin \theta \zeta_2, x_2 = \sin \theta \zeta_1 + \cos \theta \zeta_2, x_3 = \zeta_3\).
Pure stretching in axial direction yields a diagonal deformation gradient \(F = diag(\alpha_1,\alpha_2,\alpha_3)\).
Example: \(x_1 = \alpha_1 \zeta_1, x_2 = \alpha_2 \zeta_2, x_3 = \alpha_3 \zeta_3\)
Pure shear yields a symmetric deformation gradient \(F^T = F\).
6.3 Polar decomposition of \(F\)
In order to decompose \(F\) systematically, we introduce the polar decomposition. Our aim is to systematically characterize the deformation mapping \(\phi\) based on our knowledge of the deformation gradient \(F\).
\(F\) can be uniquely decomposed into
\[ F = Q \cdot U \qquad \text{or alternatively} \qquad F = V \cdot Q. \]
Here, \(Q\) is an orthogonal matrix and \(U\) and \(V\) are symmetric, positive definite matrices respectively. If we interpret \(F\) as a linear operator this means that \(F\) is decomposed into subsequently applied operators \(U\) and \(Q\), or \(Q\) and \(V\).
Proof (the spd-square-root step is treated in detail in (Gonzalez and Stuart 2001)): We define
\[ U := (F^T F)^{\tfrac{1}{2}} \qquad \text{and} \qquad Q:= F U^{-1}, \]
and deduce the following:
- \(F^T F\) is symmetric
- \(\mathbf x^T F^T F \mathbf x = (F \mathbf x)^T F \mathbf x > 0\) for all \(\mathbf x\).
- Hence, \(F^T F\) is spd and has a full set of eigenvalues. It can be diagonalized and square rooted.
- As a consequence, \(U\) exists and is invertible, such that also \(Q\) exists.
- Finally, we can show that \(Q\) is orthogonal (\(Q^T Q = I\)), which completes the proof.
The same can be shown for
\[ V := (F F^T)^{\tfrac{1}{2}} \qquad \text{and} \qquad Q:= V^{-1} F, \]
\(U\) and \(V\) appearing in the right and left polar decomposition of \(F\) code information on the strain and are referred to as the right and left stretch tensors. They are not identical in general, but share the same eigenvalues, referred to as the principle stretches. The sets of eigenvectors \(\mathbf u_i\) and \(\mathbf v_i\) are called the right and left principle stretch directions.
The deformation gradient’s polar decomposition can be physically interpreted as subsequent rotation \(Q\) and stretch \(U\) and \(V\) or subsequent stretch and rotation respectively. We get the following transformation rules for a material particle \(\Delta \mathbf \zeta\):
\[ \begin{aligned} \Delta \mathbf x &= F \Delta \mathbf \zeta = Q ( U \mathbf \Delta \zeta)\\ \Delta \mathbf x &= F \Delta \mathbf \zeta = V ( Q \mathbf \Delta \zeta). \end{aligned} \]
There are other relevant strain tensors. For example:
\(B : = V^2 = F F^T\): left Cauchy-Green tensor
\(C : = U^2 = F^T F\): right Cauchy-Green tensor
Note, that these have been used for the construction of \(U\) and \(V\).
Rigid body deformation means vanishing strain: \[U=V=B=C=I.\]
Volume preserving deformation is characterized by \(\text{det}(F) = 1\), because
\[ \int_{\Omega_{\mathbf x}} d \mathbf x = \int_{\Omega_{\mathbf \zeta}} \underbrace{\left| \tfrac{\partial \mathbf x}{\partial \mathbf \zeta}\right|}_{\mathclap{\substack{{\text{det}(F) \,:\, \text{Jacobi}} \\ \text{determinant of the} \\ \text{deformation gradient}}}} d \mathbf \zeta. \]
With that, we can single out isotropic dilation and volume preserving strain and yield:
\[ F= \underbrace{Q}_{\mathclap{\text{rotation}}} \cdot \underbrace{\text{det}(F) I }_{\substack{\mathclap{\text{isotropic}} \\ \mathclap{\text{dilation}}}} \cdot \underbrace{\dfrac{1}{\text{det}(F)} U}_{\substack{\mathclap{\text{volume}} \\ \mathclap{\text{preserving strain}}}}. \]
This means that locally any deformation can be understood as subsequent volume preserving stretching, isotropic dilation and rotation.
6.4 General material theory in a nutshell
In elasticity theory we are interested in materials, whose stress state depends on (local) strain alone. This can be further formalized as follows:
6.4.1 Elasticity
We call a solid material elastic, if its constitutive law \(\sigma = \hat \sigma (F)\) depends only on the deformation gradient \(F\).
6.4.2 Symmetry
If we transform the material’s reference configuration according to
\[\mathbf \zeta \to \mathbf \eta ( \mathbf \zeta ),\]
we get
\[ F_{ij}^{(\eta)} = \tfrac{\partial \phi_i}{\partial \eta_j} = \tfrac{\partial \phi_i}{\partial \zeta_k} \tfrac{\partial \zeta_k}{\partial \eta_j} = F_{ik}^{(\zeta)} \underbrace{P_{kj}}_{\mathclap{\text{transformation matrix}}} \qquad \Leftrightarrow \qquad F{(\eta)} = F{(\zeta)} \cdot P \]
The material is referred to as symmetric with respect to \(P\), if the result of the constitutive relation \(\hat \sigma (*)\) is unaffected by the change of the reference configuration, hence
\[ \hat \sigma^{(\zeta)} (*) = \hat \sigma^{(\eta)} (* P). \]
6.4.3 Isotropy
A material that is symmetric in the above sense for all \(P=Q\) with \(Q\) orthogonal is called isotropic. Note, that if we combine this with the former definition of elasticity, isotropy implies
\[ \hat \sigma (F) \underbrace{=}_{\text{polar decomposition}} \hat \sigma ( V \cdot Q) \underbrace{=}_{\text{symmetry}} \hat \sigma ( V \cdot Q \cdot P) \underbrace{=}_{\text{isotropy: pick } P = Q^T} \hat \sigma ( V). \]
This means, that in an isotropic, elastic material, the constitutive relation \(\hat \sigma (*)\) is a function of rotation-less strain alone!
6.4.4 Objectivity
Objectivity (also called material frame indifference) states that the constitutive relation is invariant under rigid Euclidean transformations of the spatial frame. Two observers in rigidly rotating frames must measure the same material law.
To make this concrete, consider a Euclidean transformation of the deformed configuration
\[ \mathbf y = Q(t) \mathbf x + \mathbf b(t), \]
in which \(Q(t)\) is orthogonal (a potentially time-dependent rotation) and \(\mathbf b(t)\) is a (potentially time-dependent) shift. Under this change of observer:
\[ F^{(\mathbf y)} = Q \cdot F^{(\mathbf x)}, \qquad \sigma^{(\mathbf y)} = Q \cdot \sigma^{(\mathbf x)} \cdot Q^T, \]
since rotating the spatial frame multiplies \(F\) from the left and conjugates the Cauchy stress (which is itself a spatial tensor). Objectivity now demands that the constitutive function \(\hat \sigma\) — being a property of the material, not of the observer — is the same in both frames, \(\hat\sigma^{(\mathbf y)} = \hat\sigma^{(\mathbf x)} =: \hat\sigma\). Combined with the transformation laws above, this forces the explicit constraint
\[ \hat \sigma (Q \cdot F) \;=\; Q \cdot \hat \sigma (F) \cdot Q^T \qquad \forall \, Q \in O(3). \tag{6.1}\]
This is the actual mathematical content of objectivity: a family of transformation rules on \(\hat\sigma\), one for each orthogonal \(Q\). Note that, in contrast to symmetry and isotropy — which act on the reference frame by right-multiplication, \(F \mapsto F P\) — objectivity acts on the spatial frame by left-multiplication and output conjugation, \(F \mapsto Q F\) and \(\hat\sigma \mapsto Q \hat\sigma Q^T\).
A direct consequence: take the right polar decomposition \(F = Q \cdot U\) (with the same rotation \(Q\) that appears in the left polar \(F = V Q\), see §6.3) and apply Equation 6.1 with the spatial rotation chosen as \(Q^T\). Then
\[ \hat\sigma(Q^T \cdot Q \cdot U) \;=\; \hat\sigma(U) \;=\; Q^T \cdot \hat\sigma(F) \cdot Q \qquad \Longleftrightarrow \qquad \hat\sigma(F) \;=\; Q \cdot \hat\sigma(U) \cdot Q^T. \]
Objectivity alone therefore says: \(\hat\sigma\) on a generic \(F\) is determined by \(\hat\sigma\) on the symmetric stretch \(U\) (equivalently, on the right Cauchy-Green tensor \(C = U^2\)), up to the known rotation \(Q\). The 9-component dependence on \(F\) has been reduced to 6 components.
6.4.5 Hyperelasticity
Elasticity says the stress is a function of the deformation gradient. A thermodynamic strengthening of this — required, in fact, to rule out perpetual-motion behaviour as we will see in §6.7 — is the following.
A material is called hyperelastic if there exists a scalar strain-energy density \(W\) — elastic energy stored per unit reference volume, a function of strain only — from which the stress derives by differentiation. The precise form of the differentiation depends on which strain measure \(W\) is written in; at the small-strain level relevant for §6.7, we will write it as \(\sigma_{ij} = \partial W / \partial D_{ij}\).
Hyperelasticity ensures that the work done by internal stresses along any closed deformation cycle vanishes — i.e., elastic energy is path-independent. Essentially every elastic model used in practice (linear, neo-Hookean, St-Venant–Kirchhoff, Mooney–Rivlin) is hyperelastic. The strain-energy density was introduced into elasticity theory by Green in 1839 (Maugin 2016, vol. 223, sec. 1.2.7); we will put it to work in §6.7 to obtain the major symmetry of the elasticity 4-tensor.
The Isotropy chain above delivered \(\hat\sigma(F) = \hat\sigma(V)\) from elasticity + isotropy alone — no objectivity was used. What objectivity adds, on top of this equation, is a conjugation rule on the function \(\hat\sigma\) itself.
To derive it, observe that under the spatial rotation \(F \mapsto Q F\), the left stretch transforms as \(V \mapsto Q V Q^T\) (this is just \(V_{QF} = ((QF)(QF)^T)^{1/2} = (Q V^2 Q^T)^{1/2} = Q V Q^T\), using that the symmetric pd square root is unique and commutes with orthogonal conjugation). Substituting \(\hat\sigma(F) = \hat\sigma(V)\) into the objectivity constraint Equation 6.1 then gives
\[ \hat\sigma(Q V Q^T) \;=\; Q \cdot \hat\sigma(V) \cdot Q^T \qquad \forall \, Q \in O(3). \tag{6.2}\]
This says \(\hat\sigma\) commutes with conjugation by orthogonal matrices — i.e., \(\hat\sigma\) is itself an isotropic tensor-valued function on symmetric positive-definite matrices. By the Rivlin–Ericksen representation theorem, the only such functions take the form
\[ \hat\sigma(V) \;=\; \alpha_0(I_1, I_2, I_3) \mathbf I \;+\; \alpha_1(I_1, I_2, I_3)\, V \;+\; \alpha_2(I_1, I_2, I_3)\, V^2, \]
where \(I_1, I_2, I_3\) are the principal invariants of \(V\) — three scalar functions of three scalar invariants. The spatial rotations have been eaten by the constraint and do not appear in the resulting expression.
The §6.7 derivation works with the linearisation of Equation 6.2 around \(V = \mathbf I\): it forces the elasticity 4-tensor \(\mathcal C\) to be an isotropic 4-tensor, which (after the additional symmetries from hyperelasticity) collapses to two material constants \(\lambda\) and \(\mu\).
6.5 Displacement kinematics
After a load has been applied, a material particle at position \(\mathbf \zeta\) in reference configuration \(B\) is displaced according to
\[ \mathbf w ( \mathbf \zeta) := \mathbf x - \mathbf \zeta = \phi ( \mathbf \zeta,t) - \mathbf \zeta,\]
which is referred to as the displacement field. Partial differentiation of the displacement field with respect to the material coordinate \(\mathbf \zeta\) yields the (material) displacement gradient
\[ \underbrace{H}_{\substack{\mathclap{\text{displacement}} \\ \mathclap{\text{gradient}}}} : = \nabla_{\mathbf \zeta} \phi(\mathbf x) - I = \tfrac{\partial \phi_i}{\partial \zeta_j} - \delta_{ij} = \underbrace{F}_{\substack{\mathclap{\text{deformation}} \\ \mathclap{\text{gradient}}}} - I \]
We can also solve this for the deformation gradient, hence \(F = H + I\).
Now recall our previous result regarding the deformation gradient’s polar decomposition, when we singled out strain from rotations! In order to construct the strains, e.g. \(V\), we first had to compute Cauchy-Green tensors. Let’s look at the left Cauchy-Green tensor:
\[ \begin{aligned} B_{ij} \;=\; F_{ik}\,F_{jk} &\;=\; \left( \delta_{ik} + \frac{\partial w_i}{\partial \zeta_k} \right) \left( \delta_{jk} + \frac{\partial w_j}{\partial \zeta_k} \right) \\[0.3em] &\;=\; \underbrace{\delta_{ij} + \frac{\partial w_i}{\partial \zeta_j} + \frac{\partial w_j}{\partial \zeta_i}}_{\text{linear contribution: } I + H + H^T} \;+\; \underbrace{\frac{\partial w_i}{\partial \zeta_k}\,\frac{\partial w_j}{\partial \zeta_k}}_{\substack{\mathclap{\text{non-linear}} \\ \mathclap{\text{contribution}}}}. \end{aligned} \]
This means that the Cauchy-Green tensor consists of a constant part, a part that is linear in the displacement gradient and further non-linear contributions:
\[ B = F \cdot F^T = I + H + H^T + \text{non-linear contributions}. \]
This fact explains the complicated (because typically non-linear) nature of general solid elasticity models. Here, however, we want to follow the idea of Hooke’s law focus on material behaviour that is characterized by very small displacements. This simplifies the situation significantly. as discussed in the next section.
6.6 Small displacement fields
The argument in this section is the first of two asymptotic reductions in this chapter (the second — geometric — reduction follows in §6.11). For a systematic treatment of the perturbation methodology used here we refer to (Holmes 2013).
We assume that the displacement variation within the material is very small and formalize this via \(\tfrac{\partial w_i}{\partial \zeta_j} = \mathcal O(\epsilon)\) with \(\epsilon \ll 1\). As a consequence to displacements varying only slightly, we find that the deformation gradient \(F\) varies only slightly from identity
\[
F = I + \mathcal O(\epsilon).
\]
Let’s furthermore assume that we want to keep all terms that scale with \(\mathcal O(\epsilon)\), yet want to neglect terms that scale with \(\mathcal O(\epsilon^2)\). The latter assumption is referred to as the small displacement assumption. The small displacement assumption leads to a significant simplifications:
First, we find that up to order \(\mathcal O(\epsilon)\) there is no difference between a differentiation of the displacement field with respect to coordinates of the reference frame \(x_i\) or material coordinates (\(\zeta_i\)):
\[ \begin{aligned} \frac{\partial w_i}{\partial \zeta_j} = \frac{\partial w_i}{\partial \phi_k} \frac{\partial \phi_k}{\partial \zeta_j} = \frac{\partial w_i}{\partial x_k} F_{kj} = \frac{\partial w_i}{\partial x_k} \left( \delta_{kj} + \frac{\partial w_k}{\partial \zeta_j}\right) = \frac{\partial w_i}{\partial x_j} + \mathcal O(\epsilon^2). \end{aligned} \]
This means basically, that the difference between a Eulerian and Lagrangian description of displacement field is of order \(\mathcal O(\epsilon^2)\) and can be neglected.
We furthermore find that left and right Cauchy-Green tensor are identical with vanishing non-linear contributions \[ \begin{aligned} B = C = I + H + H^T + \mathcal O(\epsilon^2), \end{aligned} \]
which implies that \(U\) and \(V\) are also identical given by \[ \begin{aligned} U = V = I + \frac{1}{2} (H + H^T) + \mathcal O(\epsilon^2), \end{aligned} \]
as one verifies by computing \(U^2 = (I + \tfrac{1}{2}(H+H^T))^2 = I + (H + H^T) + \mathcal O(\epsilon^2) = C\), matching the linearised right Cauchy-Green tensor (and \(V^2 = B\) analogously).
The inverse stretch tensors accordingly simplify into \[ \begin{aligned} U^{-1} = V^{-1} = I - \frac{1}{2} \left( H + H^T \right) + \mathcal O(\epsilon^2). \end{aligned} \]
Finally, the orthogonal rotation tensor can be determined to be \[ \begin{aligned} Q = I + \frac{1}{2} \left( H - H^T \right) + \mathcal O(\epsilon^2). \end{aligned} \]
To verify, recall \(F = Q\,U\). Substituting both linearised tensors: \[ \begin{aligned} Q \cdot U &\;=\; \left(I + \tfrac{1}{2}(H - H^T)\right)\left(I + \tfrac{1}{2}(H + H^T)\right) \\[0.3em] &\;=\; I + \tfrac{1}{2}(H+H^T) + \tfrac{1}{2}(H-H^T) + \mathcal O(\epsilon^2) \;=\; I + H + \mathcal O(\epsilon^2) \;=\; F. \end{aligned} \]
We conclude that for small displacements, the polar decomposition is given as \[ \begin{aligned} F = \underbrace{I + \frac{1}{2} \left( H + H^T \right)}_{\substack{\mathclap{\text{corresponds to}} \\ \mathclap{\text{strain } V}}} + \underbrace{\frac{1}{2} \left(H - H^T \right)}_{\substack{\mathclap{\text{corresponds to}} \\ \mathclap{\text{rotation } Q}}} + \mathcal O(\epsilon^2). \end{aligned} \]
The symmetric tensor \(\frac{1}{2} \left( H + H^T \right)\) is also referred to as the small strain tensor. We find that
Rigid body deformation yields \(B=C=I\), hence \(\tfrac{1}{2} \left( H + H^T \right)=0\)
Rotation is controlled by \(\tfrac{1}{2} \left( H - H^T \right)=0\)
Volume preservation implies \(\text{det}(F) = 1\), hence \(\tfrac{\partial w_k}{\partial x_k}=0\)
which reminds us of findings, when we decomposed the velocity vector (see Section 4.2).
Two distinct smallness conditions appear in the linear theory, and they have different consequences:
- Small displacement (this section): the full displacement gradient is small, \(\partial w_i / \partial \zeta_j = \mathcal O(\epsilon)\). This is what collapses Eulerian and Lagrangian descriptions to leading order, replaces \(F\) by \(I + H\), and identifies \(V \approx I + D\) with \(D\) the small-strain tensor.
- Small strain: only the symmetric part is small, \(D_{ij} = \mathcal O(\epsilon)\). This is the weaker condition — and the only one needed for the constitutive linearisation in §6.7.
A body can rotate near-rigidly through a large angle while staying at small strain; in that regime the constitutive law (§6.7) remains valid, but the kinematic linearisation \(F \approx I + H\) does not — the rotation must be retained nonlinearly. This is the distinction between mechanical linearity (linear stress-strain law) and geometric linearity (small displacement gradient). The Kirchhoff–Love plate of §6.11 lives in the strictly linear regime; the Föppl–von Kármán model (cf. §6.11, Edge of the linear theory) keeps mechanical linearity but retains geometric nonlinearity to capture buckling.
6.7 Continuous formulation of Hooke’s law
We are now in a position to derive — not postulate — the continuous form of Hooke’s law from the apparatus we already have. The chain has four steps.
Setup. From §6.4 we know that for an isotropic, elastic, objective material, the stress depends only on the rotation-less strain, \(\hat\sigma = \hat\sigma(V)\). From §6.6 we know that under the small-displacement assumption, \[ V \;=\; I + D + \mathcal O(\epsilon^2), \qquad D := \tfrac{1}{2}(H + H^T). \]
Step 1 — Taylor expansion. Assuming \(\hat\sigma\) smooth around the reference configuration \(V = I\) and small displacements, expand: \[ \hat\sigma(I + D) \;=\; \hat\sigma(I) \;+\; \mathcal C : D \;+\; \mathcal O(|D|^2), \] where \(\mathcal C\) is the derivative of \(\hat\sigma\) at the reference state — a fourth-order tensor with components \[ C_{ijkl} \;:=\; \left.\frac{\partial \hat\sigma_{ij}}{\partial D_{kl}}\right|_{D = 0}, \qquad \sigma_{ij} \;=\; C_{ijkl}\,D_{kl}\;\text{ at leading order}. \] \(\mathcal C\) is called the elasticity tensor. It encodes the linear map \(D \mapsto \sigma\) — not the value of \(\sigma\) at any particular state. Specifying a linear map between two 6-dim spaces (symmetric 2-tensors) requires \(6 \times 6 = 36\) entries; raw counting of components gives \(3^4 = 81\).
Step 2 — Stress-free reference. We add the physical postulate that the reference configuration is unstressed, \[ \hat\sigma(I) \;=\; 0. \] At leading order in \(\epsilon\) this leaves \[ \sigma_{ij} \;=\; C_{ijkl}\,D_{kl}. \tag{6.3}\] The constitutive law is linear in the strain. We have not yet fixed the form of \(\mathcal C\).
Step 3 — Symmetries of \(\mathcal C\). We pin down two minor index symmetries from prior chapters and one major symmetry from a thermodynamic postulate.
Minor symmetry 1 — from \(\sigma\) symmetric. Balance of angular momentum (§5) forces \(\sigma_{ij} = \sigma_{ji}\). Inserting this into Equation 6.3 and swapping \(i\leftrightarrow j\): \[ \sigma_{ij} = C_{ijkl}\,D_{kl} \;=\; \sigma_{ji} = C_{jikl}\,D_{kl} \quad\forall D \quad\Longrightarrow\quad C_{ijkl} = C_{jikl}. \]
Minor symmetry 2 — from \(D\) symmetric. By construction \(D_{kl} = D_{lk}\), so only the \((kl)\)-symmetric part of \(\mathcal C\) contributes to the contraction: \[ C_{ijkl}\,D_{kl} \;=\; \tfrac{1}{2}\!\left(C_{ijkl} + C_{ijlk}\right) D_{kl} \quad\Longrightarrow\quad C_{ijkl} = C_{ijlk}. \]
Counting after minor symmetries (81 → 36). A symmetric 3×3 tensor has 6 independent components (3 diagonal + 3 off-diagonal). After the two minor symmetries, \(\mathcal C\) is the matrix of a linear map between two 6-dimensional spaces of symmetric tensors. Such a map is encoded by a \(6\times 6\) matrix \(\bar{\mathcal C}_{IJ}\) via Voigt notation — a single index \(I \in \{1,\dots,6\}\) enumerating the symmetric pair \((ij)\): \[ \sigma_{ij} = C_{ijkl}\,D_{kl} \quad\longleftrightarrow\quad \bar\sigma_I = \bar{\mathcal C}_{IJ}\,\bar D_J. \] The \(6 \times 6 = 36\) entries of \(\bar{\mathcal C}\) are exactly the 36 independent components of \(\mathcal C\) remaining after the two minor symmetries.
The major symmetry \(C_{ijkl} = C_{klij}\) may look like an aesthetic refinement. It is in fact a thermodynamic necessity — without it, an elastic material can be cycled to extract net mechanical work from nothing.
A sharp construction in the fully nonlinear setting is given in (Howell, Kozyreff, and Ockendon 2009, vol. 43, sec. 5.3.4): the constitutive law \(S = (\det C - 1)\, \mathbf I\) (with \(S\) the second Piola-Kirchhoff stress and \(C\) the right Cauchy-Green tensor) is isotropic, elastic, and stress-free at the reference state — every axiom of §6.4 holds. Yet under a suitably chosen periodic deformation history the rate of stored elastic energy integrates to a non-zero net value over a closed cycle, the sign depending on the cycle parameters. A material satisfying this law would be a perpetual-motion machine: drive it through the cycle, extract net mechanical work, repeat. The pathology is that the work along a closed loop in deformation space depends on the path, not just on the endpoints.
The cure is to demand that the work is path-independent — equivalently, that there is a scalar potential from which the stress derives. This is exactly the hyperelastic postulate of §6.4. As we show next, Schwarz’s theorem on the resulting potential then yields the major symmetry as a consequence, rather than as a separately motivated axiom.
Major symmetry — from hyperelasticity. The hyperelastic postulate of §6.4 supplies precisely the missing ingredient. At the small-strain level, it asserts the existence of a scalar strain-energy density \(W(D)\) such that \[ \sigma_{ij} \;=\; \frac{\partial W}{\partial D_{ij}}. \tag{6.4}\] We adopt Equation 6.4 going forward.
Inserting Equation 6.4 into the definition of \(\mathcal C\) and applying Schwarz’s theorem on the equality of mixed partials (valid for \(W \in C^2\)): \[ C_{ijkl} \;=\; \frac{\partial \sigma_{ij}}{\partial D_{kl}} \;=\; \frac{\partial^2 W}{\partial D_{kl}\,\partial D_{ij}} \;\overset{\text{Schwarz}}{=}\; \frac{\partial^2 W}{\partial D_{ij}\,\partial D_{kl}} \;=\; C_{klij}. \] This is the major symmetry \(C_{ijkl} = C_{klij}\).
Counting after the major symmetry (36 → 21). In the Voigt picture, the major symmetry translates directly into \(\bar{\mathcal C}_{IJ} = \bar{\mathcal C}_{JI}\): the \(6\times 6\) matrix is symmetric. A symmetric \(6\times 6\) matrix has \[ \binom{6+1}{2} \;=\; \frac{6 \cdot 7}{2} \;=\; 21 \] independent entries (6 on the diagonal + 15 above it). Hence 36 → 21. The 21-component object is the most general fully anisotropic hyperelastic linear material — it is the starting point for the standard catalogue of crystal symmetry classes (triclinic 21, monoclinic 13, orthotropic 9, cubic 3, isotropic 2, …).
Step 4 — Isotropy and the representation theorem. The §6.4 isotropy pillar requires \(\mathcal C\) to be an isotropic 4-tensor — invariant under all rotations of the reference frame: \[ Q_{i'i}\,Q_{j'j}\,Q_{k'k}\,Q_{l'l}\,C_{ijkl} \;=\; C_{i'j'k'l'} \qquad \forall\,Q\in SO(3). \] The representation theorem for isotropic 4-tensors states that the only building blocks of an isotropic 4-tensor are products of Kronecker deltas, giving the most general form \[ C_{ijkl} \;=\; a\,\delta_{ij}\delta_{kl} \;+\; b\,\delta_{ik}\delta_{jl} \;+\; c\,\delta_{il}\delta_{jk} \] with three free scalars. (Any other index combination reduces to one of these via the symmetry \(\delta_{ij} = \delta_{ji}\).) Imposing the minor symmetry \(C_{ijkl} = C_{ijlk}\) swaps \(k \leftrightarrow l\) in the second and third terms — these are exchanged — so their coefficients must agree, \(b = c\).
Setting \(\lambda := a\), \(\mu := b = c\) yields exactly two free scalars: \[ C_{ijkl} \;=\; \lambda\,\delta_{ij}\delta_{kl} \;+\; \mu\,(\delta_{ik}\delta_{jl} + \delta_{il}\delta_{jk}). \tag{6.5}\]
Substituting Equation 6.5 into Equation 6.3 and using \(D_{kl} = D_{lk}\): \[ \sigma_{ij} \;=\; \lambda\,\delta_{ij}\,\underbrace{\delta_{kl} D_{kl}}_{= D_{kk} = tr(D)} \;+\; \mu\,\big(\underbrace{\delta_{ik}\delta_{jl} D_{kl}}_{= D_{ij}} + \underbrace{\delta_{il}\delta_{jk} D_{kl}}_{= D_{ji} = D_{ij}}\big), \] yielding the continuous form of Hooke’s law: \[ \boxed{\;\sigma \;=\; \lambda\,tr(D)\,\mathbf I \;+\; 2\mu\,D.\;} \tag{6.6}\] The factor 2 on the shear term comes from the symmetric pair of \(\delta\delta\)-products in Equation 6.5; this is exactly why \(\mu\) is conventionally called the shear modulus rather than half the shear modulus.
The constants \(\lambda\) and \(\mu\) are the Lamé constants, measurable for each material. The trace \(tr(D) = \partial_k w_k\) codes the material’s local change in volume.
The four structural ingredients of the derivation reduce the elasticity 4-tensor \(\mathcal C\) in distinct steps. Summarising:
| Step | Axiom / restriction | Constraint on \(\mathcal C_{ijkl}\) | Independent components |
|---|---|---|---|
| 1 | Linearisation of \(\hat\sigma(V)\) at \(V = \mathbf I\) | — | 81 |
| 2 | Stress-free reference, \(\hat\sigma(\mathbf I) = 0\) | — | 81 |
| 3a | \(\sigma\) symmetric (angular momentum) | \(C_{ijkl} = C_{jikl}\) | 54 |
| 3b | \(D\) symmetric | \(C_{ijkl} = C_{ijlk}\) | 36 |
| 3c | Hyperelasticity (existence of \(W\), Schwarz) | \(C_{ijkl} = C_{klij}\) | 21 |
| 4 | Isotropy + objectivity (representation theorem) | \(Q_{i'i}Q_{j'j}Q_{k'k}Q_{l'l} C_{ijkl} = C_{i'j'k'l'}\) | 2 (\(\lambda, \mu\)) |
The 21-component layer (anisotropic hyperelastic) is the starting point for the standard catalogue of crystal symmetry classes (triclinic 21, monoclinic 13, orthotropic 9, cubic 3, isotropic 2, …). The 2-component endpoint of step 4 is what makes Hooke’s law tractable.
The representation theorem yields exactly two scalar parameters. All further constants you will meet — \(K\) (bulk modulus, §6.7), \(E\) (Young’s modulus, §6.1), \(\nu\) (Poisson’s ratio, §6.1), \(G \equiv \mu\) (shear modulus) — are derived combinations of \(\lambda\) and \(\mu\), not independent parameters. Any two of \(\{\lambda, \mu, K, E, \nu, G\}\) determine the other four.
Hooke’s law can alternatively be stated in terms of the small-strain deviator
\[D^{(d)} := D - \frac{1}{3} tr (D)I,\]
for which \(tr (D^{(d)}) = 0\). This yields
\[ \begin{aligned} \sigma &= 2 \mu D^{(d)} + K tr (D) I, \end{aligned} \]
with \(K= \lambda + \tfrac{2}{3} \mu\). In index notation, this reads
\[ \begin{aligned} \sigma_{ij} &= 2 \mu \left( D_{ij} - \frac{1}{3} D_{kk} \delta_{ij} \right) + K D_{kk} \delta_{ij}. \end{aligned} \tag{6.7}\]
The trace can be computed via convolution (setting \(i=j\) and summation), which results in
\[ \sigma_{kk} = 3 K D_{kk} \qquad \Leftrightarrow \qquad tr(\sigma) = 3 K tr(D). \tag{6.8}\]
Hence, the trace of the stress tensor and the trace of the small-strain tensor are proportional to each other.
Let’s for instance consider a hydrostatic situation (lithostatic/ glaciostatic), in which \(\sigma = - p I\), hence \(tr(\sigma) = - 3 p\) and consequently
\[ tr(D) = - \frac{p}{K}. \]
That means that the volume strain \(tr(D)\) is proportional to pressure p with proportionality constant \(-K^{-1}\). \(K\) is therefore called the volume compression modulus.
For shear stresses, we get \(\delta_{ij} = 0\), hence
\[ \sigma_{ij} = 2 \mu D_{ij} = \mu ( \partial_j w_i + \partial_i w_j). \]
Parameter \(\mu\) is hence referred to as the shear modulus.
6.7.1 Consistency check: matching the rod experiment
We started this chapter (§6.1) from classical Hooke’s law for a longitudinal rod, written in terms of Young’s modulus \(E\) and Poisson’s ratio \(\nu\) — two phenomenological constants. Step 4 above predicts two independent Lamé constants \(\lambda\) and \(\mu\) from purely structural arguments. The two parameter counts agree. We now perform the consistency check by computing the rod response from Equation 6.6 and comparing to §6.1.
We solve Equation 6.7 for the small-strain \(D\) by first substituting in Equation 6.8
\[ \sigma_{ij} = 2 \mu \left( D_{ij} - \frac{1}{9K} \sigma_{kk} \delta_{ij} \right) + \frac{1}{3} \sigma_{kk} \delta_{ij}. \]
Now, it is possible to solve for \(D_{ij}\), which yields
\[ D_{ij} = \frac{1}{9K} \sigma_{kk} \delta_{ij} + \frac{1}{2 \mu} ( \sigma_{ij} - \frac{1}{3} \sigma_{kk} \delta_{ij} ). \]
In our initial strained rod example, we had \(D_{11} = \tfrac{\Delta l}{l}\) and \(D_{22} = \tfrac{\Delta S}{S}\). Axial load was exerted in \(x_1\) direction, hence \(\sigma_{11} = \tfrac{P}{S}\). It was zero elsewhere. This resolves in
\[ \begin{aligned} D_{11} = \frac{\Delta l}{l} &= \frac{1}{9K} \frac{P}{S} + \frac{1}{2 \mu} \left( \frac{P}{S} -\frac{1}{3}\frac{P}{S} \right) \\ &= \left( \frac{1}{9K} + \frac{1}{3 \mu} \right) \frac{P}{S} \overset{!}{=} \frac{1}{E}\frac{P}{S} \end{aligned} \] \[ \begin{aligned} D_{22} = \frac{\Delta S}{S} &= \frac{1}{9K} \frac{P}{S} - \frac{1}{6 \mu} \frac{P}{S} \overset{!}{=} - \frac{\nu}{E}\frac{P}{S} \end{aligned} \]
We can solve these relations for \(\mu, K\), and \(\lambda\) and find
\[ \mu=\frac{E}{2(1+\nu)} \qquad K=\frac{E}{3(1-2\nu)} \qquad \lambda=\frac{\nu E}{(1+\nu)(1-2\nu)}. \]
The structural derivation produces exactly the same number of constants (\(\lambda\) and \(\mu\), equivalently \(\{K,\mu\}\), equivalently \(\{E,\nu\}\)) as the rod experiment did — and it tells us how the engineering constants \(E\) and \(\nu\) are encoded in the structural pair \(\lambda, \mu\). This closes the loop between the experimental motivation of §6.1 and the structural derivation that opens §6.7.
6.8 Theory of linear elasticity
The major goal of this chapter is to derive a mathematical process model, hence a PDE system, that allows us to compute the spatio-temporal evolution of the displacement field \(\mathbf w\). An alternative derivation route via the principle of virtual work is given in (Howell, Kozyreff, and Ockendon 2009 ch. 4).
We start from the momentum balance (body forces are neglecting for time being):
\[ \rho \frac{D}{Dt} \mathbf v = \nabla \cdot \sigma \]
First, we rewrite total derivative in Lagrangian coordinates
\[\frac{D}{Dt} \mathbf v = \frac{\partial}{\partial t} \mathbf{\tilde v},\]
which allows us to substitute in the deformation mapping and displacement field:
\[ \frac{\partial}{\partial t} \underbrace{\mathbf{\tilde v}(\zeta,t)}_{\substack{\mathclap{\text{velocity in}} \\ \mathclap{\text{Lagrangian reference}}}} = \frac{\partial}{\partial t} \left(\frac{\partial}{\partial t} \phi(\zeta,t)\right) = \frac{\partial^2}{\partial t^2} \left( \mathbf{w}(\zeta,t) + \zeta \right) = \frac{\partial^2}{\partial t^2} \mathbf{w}(\zeta,t). \]
This finally yields
\[ \rho \frac{\partial^2}{\partial t^2} \mathbf w = \nabla \cdot \sigma. \]
We can now substitute in Hooke’s law in continuous materials: \[ \rho \frac{\partial^2}{\partial t^2} \mathbf w = 2 \mu \nabla \cdot D + \lambda \nabla \cdot (tr(D) I). \]
in terms of small strain tensor \(D = \tfrac{1}{2} \left( H + H^T\right)\). It’s divergence is given by
\[ \begin{aligned} \nabla \cdot D &= \frac{1}{2} \nabla \cdot \left( H + H^T \right) \\ &= \frac{1}{2} \nabla \cdot \left( \nabla \mathbf w + (\nabla \mathbf w)^T \right) = \frac{1}{2} \left( \nabla (\nabla \cdot \mathbf w) + \Delta \mathbf w \right). \end{aligned} \]
Exploiting the operator identity
\[ \nabla \cdot ( tr(D) I) = \nabla (\nabla \cdot \mathbf w) \]
finally gets us the dynamic theory of linear elasticity
\[ \rho \frac{\partial^2}{\partial t^2} \mathbf w = \mu \Delta \mathbf w + \frac{\mu}{1-2 \nu} \nabla (\nabla \cdot \mathbf w). \]
With the vector identity \(\nabla (\nabla \cdot \mathbf w) = \Delta \mathbf w + \nabla \times \nabla \times \mathbf w\), this can alternatively be written as
\[ \rho \frac{\partial^2}{\partial t^2} \mathbf w = \frac{\mu}{1-2\nu} \left( 2 ( 1-\nu) \Delta \mathbf w + \nabla \times \nabla \times \mathbf w \right). \]
6.9 Static theory of elasticity
The static theory of elasticity goes back to Navier (1827); see also (Howell, Kozyreff, and Ockendon 2009 ch. 4) for a treatment of boundary value problems and standard analytical solutions. It results from the dynamic theory by neglecting transient effects, hence
\[\frac{\partial^2}{\partial t^2} \mathbf w = 0,\]
which yields
\[ 0 = ( 1- 2 \nu) \Delta \mathbf w + \nabla (\nabla \cdot \mathbf w). \]
The dynamic Navier equation carried both the density \(\rho\) and the elastic moduli \(\mu, \nu\) as independent material constants. The static reduction collapses this to a single material parameter: Poisson’s ratio \(\nu\). Dividing through by \(\mu/(1-2\nu)\) absorbed the shear modulus into the right-hand side, and dropping the inertial term removed \(\rho\) from the field equation. The traction boundary condition \(\sigma \cdot \mathbf n = \mathbf g\) is then implicitly understood with \(\mathbf g\) scaled by \(\mu\) — equivalently, all stresses are non-dimensionalised by the shear modulus before solving. In the absence of body forces, the static elasticity field problem is parametrised entirely by \(\nu\).
The static theory has important properties:
The divergence of the displacement field \(\nabla \cdot \mathbf w\), namely \(tr(D)\), is a harmonic function, hence a solution to the homogeneous Laplace equation \(\Delta ( \cdot ) = 0\). We see this from \[ \begin{aligned} 0 &= \nabla \cdot \left( ( 1- 2 \nu) \Delta \mathbf w + \nabla (\nabla \cdot \mathbf w) \right) \\[1em] &= \underbrace{( 1-2\nu) \Delta \nabla \cdot \mathbf w}_{\nabla \cdot \text{ and } \Delta \text{ commute}} + \underbrace{\nabla \cdot \nabla (\nabla \cdot \mathbf w) }_{= \Delta \nabla \cdot \mathbf w} \\[1em] &= 2( 1-\nu) \Delta (\nabla \cdot \mathbf w) \end{aligned} \]
The displacement field \(\mathbf w\) is biharmonic, which can be seen by applying the Laplacian operator to static elasticity theory: \[ \begin{aligned} 0 &= \Delta \left( ( 1- 2 \nu) \Delta \mathbf w + \nabla \nabla \cdot \mathbf w \right) \\[1em] &= (1-2\nu) \Delta \Delta \mathbf w + \underbrace{\Delta \nabla \nabla \cdot \mathbf w }_{=\nabla \underbrace{\Delta \nabla \cdot \mathbf w}_{=0} =0} \\ &= ( 1-2\nu) \Delta \Delta \mathbf w, \end{aligned} \] hence \(\Delta \Delta \mathbf w = 0\). The latter defines a biharmonic function.
This offers a pathway towards a numeric solution: In order to solve for the displacement field, we finally need boundary conditions, e.g.
a prescribed displacement given as \(\mathbf w = f(\mathbf x)\) for \(\mathbf x \in \partial \Omega\), or
a prescribed traction vector \(\sigma \cdot \mathbf n = f(\mathbf x)\) for \(\mathbf x \in \partial \Omega\).
6.9.1 Worked example — sinusoidal traction on a half-plane
To anchor the abstract result \(\Delta^2 \mathbf w = 0\) before moving on to plate theory, we work out one closed-form 2D problem in plane strain. Take the upper half-plane \(y > 0\) loaded on its surface \(y = 0\) with a sinusoidal normal traction of wavenumber \(k > 0\) and amplitude \(p_0\):
\[ \sigma_{yy}(x, 0) = -p_0 \cos(k x), \qquad \sigma_{xy}(x, 0) = 0, \]
and require all stresses to decay as \(y \to \infty\).
In 2D, equilibrium \(\partial_j \sigma_{ij} = 0\) without body forces is identically satisfied by introducing a scalar Airy stress function \(\varphi(x, y)\) via \(\sigma_{xx} = \partial_y^2 \varphi\), \(\sigma_{yy} = \partial_x^2 \varphi\), \(\sigma_{xy} = - \partial_x \partial_y \varphi\). Compatibility then reduces to a single scalar biharmonic equation \(\Delta^2 \varphi = 0\) — the same structural property derived above, now in the stress potential.
A decaying ansatz consistent with the surface load and the far-field condition is
\[ \varphi(x, y) = (A + B y) \, e^{-k y} \cos(k x), \]
which is biharmonic by construction. Computing the stresses and applying the BCs at \(y = 0\) pins down \(A = p_0/k^2\) and \(B = p_0/k\), giving
\[ \varphi(x, y) = \frac{p_0}{k^2}\,(1 + k y)\,e^{-k y} \cos(k x). \tag{6.9}\]
The structural reading: each Fourier mode of the surface load — wavenumber \(k\) — produces stresses with an \(e^{-|k| y}\) envelope into the bulk. Short-wavelength surface details decay first; long-wavelength components penetrate deepest. This decay is the gateway to the Saint-Venant principle below.
6.9.2 Saint-Venant principle
The exponential factor \(e^{-|k| y}\) in Equation 6.9 is a per-mode statement: each Fourier component decays on its own length scale \(1/|k|\). The descriptive statement about the system is what this implies for the superposition: at any depth \(y\), the bulk only sees surface modes with \(|k| \lesssim 1/y\) — shorter-wavelength components have already been damped by orders of magnitude. Equivalently, the half-plane acts as a depth-dependent low-pass spatial filter on the surface load, with cutoff wavenumber \(|k|_c \sim 1/y\) that shrinks as we move deeper.
In the limit \(y \to \infty\) only the \(k = 0\) component — the resultant force and moment of the surface load — survives. Two surface load distributions with identical resultants therefore produce stress fields that differ only in a thin near-surface layer. This is the Saint-Venant principle.
See Howell, Kozyreff & Ockendon for more details (Howell, Kozyreff, and Ockendon 2009, vol. 43, sec. 2.6).
6.10 Wave propagation
Wave propagation is a special case of dynamic elasticity. The argument below — Helmholtz decomposition followed by curl/divergence projections — is the standard textbook route; see (Howell, Kozyreff, and Ockendon 2009 ch. 4) for further detail and applications. Let’s decompose the displacement field into a divergence free part referred to as solenoidal part, and a curl free part referred to as the irrotational part. This type of decomposition is very famous and known as the Helmholtz decomposition: \[ \mathbf w ( \mathbf x,t) = \underbrace{\mathbf w_1(\mathbf x,t)}_{\nabla \cdot \mathbf w_1=0} + \underbrace{\mathbf w_2(\mathbf x,t)}_{\nabla \times \mathbf w_2=0} \tag{6.10}\]
Substitution into the dynamic wave equation yields
\[ \rho \frac{\partial^2}{\partial t^2} (\mathbf w_1 + \mathbf w_2) = \mu \Delta (\mathbf w_1 + \mathbf w_2) + \frac{\mu}{1 - 2 \nu} \nabla (\nabla \cdot (\mathbf w_1 + \mathbf w_2)). \]
Applying the curl operator to the latter and exploiting vector identities yields
\[ \nabla \times \underbrace{\left( \rho \frac{\partial^2}{\partial t^2} \mathbf w_1 - \mu \Delta \mathbf w_1 \right)}_{=(I)} = 0. \]
Note, that per construction of the Helmholtz decomposition also the divergence of the resulting expression vanishes: \(\nabla \cdot (I) = 0\).
We now apply the divergence operator to the decomposed wave equation. The contributions from \(\mathbf w_1\) vanish (since \(\nabla \cdot \mathbf w_1 = 0\) implies \(\nabla \cdot \Delta \mathbf w_1 = \Delta(\nabla \cdot \mathbf w_1) = 0\) as well as \(\nabla \cdot \partial_{tt} \mathbf w_1 = 0\)). For the surviving \(\mathbf w_2\) terms we use the irrotationality identity
\[ \nabla(\nabla \cdot \mathbf w_2) = \Delta \mathbf w_2 + \nabla \times \nabla \times \mathbf w_2 = \Delta \mathbf w_2, \]
which lets us combine \(\mu \Delta \mathbf w_2 + \frac{\mu}{1-2\nu} \nabla(\nabla\cdot\mathbf w_2) = \frac{2(1-\nu)}{1-2\nu}\mu\,\Delta \mathbf w_2\) into a single Laplacian term. We obtain
\[ \nabla \cdot \underbrace{\left( \rho \frac{\partial^2}{\partial t^2} \mathbf w_2 - \frac{2(1-\nu)}{1-2\nu}\mu \Delta \mathbf w_2 \right)}_{=(II)} = 0, \]
for which we similarly get \(\nabla \times (II) = 0\).
Both \((I)\) and \((II)\) have a vanishing divergence and a vanishing curl. As a consequence, both have to posses a scalar potential, which is a harmonic function \[ \begin{aligned} (I) = \nabla \phi_I \quad &\text{with}& \Delta \phi_I = 0 \\ (II) = \nabla \phi_{II} \quad &\text{with}& \Delta \phi_{II} = 0. \end{aligned} \]
For a body in which displacements vanish in the far-field, we hence conclude \(\phi_{I} = \phi_{II}=0\) to be the only solution. This in return means that both \(\mathbf w_1\), and \(\mathbf w_2\) fulfill the wave equations:
\[ \begin{aligned} \frac{\partial^2}{\partial t^2} \mathbf w_1 - a_1 \Delta \mathbf w_1 = 0 \quad \text{where} \quad &a_1 = \frac{\mu}{\rho} \\ \frac{\partial^2}{\partial t^2} \mathbf w_2 - a_2 \Delta \mathbf w_2 = 0 \quad \text{where} \quad &a_2 = \frac{2(1-\nu)}{1-2\nu} \frac{\mu}{\rho} > a_1. \end{aligned} \]
Waves of the first type have no change in volume (\(\nabla \cdot \mathbf w_1=0\)). They are characterized by distorsion of particles. Alternative names are shear waves, transverse waves, solenoidal waves or S-waves.
Waves of the second type produce volume changes and are referred to as compression expansion waves, pressure waves, longitudinal, dilational, irrotational waves or P-waves.
6.11 Dimensional reduction: plates and beams
The discussion of §§6.5–6.6 already exploited a small parameter — the kinematic \(\epsilon\) measuring the magnitude of the displacement gradient — to collapse the nonlinear continuum elasticity into the linear Navier equation. Three-dimensional Navier still describes a fully spatial PDE system in \(\mathbf w(\mathbf x, t)\). Yet many engineering structures are thin: roofs, floor slabs, ship hulls, microelectromechanical sensors, biological membranes. Their geometry suggests a second asymptotic reduction, this time geometric rather than kinematic. The methodology mirrors the lubrication argument used in the derivation of the Reynolds equation in the lubricated-flow chapter: a small geometric ratio drives an integration across the thin direction and produces a model of reduced spatial dimension.
The presentation in this section follows (Howell, Kozyreff, and Ockendon 2009 ch. 6), to which we refer for the full algebraic detail. Here, we work out the scaling argument and state the resulting plate and beam equations.
6.11.1 Motivation: a second small parameter
Consider a plate-like body that occupies a thin region
\[ \Omega = \{ (x_1, x_2, x_3) \in \mathbb R^3 \, : \, (x_1, x_2) \in S \subset \mathbb R^2, \, |x_3| \le h/2 \}, \]
with mid-surface \(S\) of in-plane characteristic length \(L\) and uniform thickness \(h\). The defining geometric assumption is
\[ \eta := \frac{h}{L} \ll 1. \]
This is the geometric small parameter — distinct from the kinematic \(\epsilon\) of §6.6, but exploited by entirely the same machinery. The two reductions are nested: we first sit inside linear elasticity (using \(\epsilon \ll 1\) to obtain Navier), and then within linear elasticity we further reduce the spatial dimensionality (using \(\eta \ll 1\)).
6.11.3 Thickness integration and moment resultants
Equilibrium of the 3D body in the thin region is then enforced integrated across the thickness. Two operations matter:
- Force resultants — integrate the equilibrium equation directly: \[ N_{\alpha\beta} := \int_{-h/2}^{h/2} \sigma_{\alpha\beta} \, dx_3. \]
- Moment resultants — multiply by \(x_3\) and integrate: \[ M_{\alpha\beta} := \int_{-h/2}^{h/2} x_3 \, \sigma_{\alpha\beta} \, dx_3. \tag{6.13}\]
Substituting the Kirchhoff–Love ansatz Equation 6.12 into the constitutive law (with the transverse normal stress \(\sigma_{33}\) neglected to leading order in \(\eta\)) and performing the \(x_3\)-integration yields a constitutive relation between bending moments and curvatures of the mid-surface,
\[ M_{\alpha\beta} = - K_b \left[ (1 - \nu) \, \partial_\alpha \partial_\beta w + \nu \, \delta_{\alpha\beta} \, \Delta_2 w \right], \tag{6.14}\]
with \(\Delta_2 := \partial_1^2 + \partial_2^2\) the 2D Laplacian on \(S\) and the plate bending stiffness
\[ K_b := \frac{E h^3}{12 (1 - \nu^2)}. \tag{6.15}\]
Note the geometric scaling \(K_b \propto h^3\): stiffness against bending is extremely sensitive to thickness. A factor-of-two thicker plate is eight times stiffer.
6.11.4 The Kirchhoff–Love plate equation
Force balance in the transverse direction, again integrated across the thickness, gives
\[ \partial_\alpha \partial_\beta M_{\alpha\beta} + q = 0, \]
where \(q(x_1, x_2)\) is the net distributed transverse load per unit mid-surface area. Substituting Equation 6.14 and using \(\partial_\alpha \partial_\beta (\partial_\alpha \partial_\beta w) = \Delta_2^2 w\) collapses the system into a single scalar PDE for the mid-surface deflection,
\[ \boxed{\quad K_b \, \Delta_2^2 \, w = q. \quad} \tag{6.16}\]
This is the Kirchhoff–Love plate equation.
We met the biharmonic operator \(\Delta^2\) once before, in the 3D static theory of §6.9, where it characterised the elastic displacement field. It now reappears, this time in 2D, as the leading-order operator of the dimensionally reduced plate model. The recurrence is not coincidental: in both cases it is a consequence of equilibrium of an isotropic linear-elastic medium — once 3D, once thickness-averaged.
The equation is fourth order in space, so each edge of \(S\) requires two boundary conditions. Common cases:
- Clamped edge — both \(w\) and \(\partial_n w\) prescribed (zero, typically).
- Simply supported edge — \(w\) and the bending moment along the edge prescribed.
- Free edge — bending moment and effective transverse shear prescribed.
For a simply supported rectangular plate \(S = [0, a] \times [0, b]\) under load \(q(x_1, x_2)\), the Navier double-Fourier-series ansatz
\[ w(x_1, x_2) = \sum_{m, n=1}^{\infty} W_{mn} \, \sin\!\left(\frac{m \pi x_1}{a}\right) \sin\!\left(\frac{n \pi x_2}{b}\right) \]
satisfies the boundary conditions term-by-term and reduces Equation 6.16 to an algebraic relation between the Fourier coefficients of \(w\) and \(q\). We do not evaluate the series here — the point is structural: the biharmonic 2D problem reduces to algebra under separation of variables, exactly as the harmonic 2D problem does.
6.11.5 Beam specialization
Restricting the geometry further so that both transverse dimensions are small — a slender beam of length \(L\), thickness \(h\), and width \(b\) with \(h, b \ll L\) — the same asymptotic argument collapses the 2D plate equation to the Euler–Bernoulli beam equation
\[ \boxed{\quad EI \, w''''(x) = q(x), \quad} \tag{6.17}\]
where \(w(x)\) is the mid-line deflection, \(q(x)\) is the load per unit length, \(E\) is Young’s modulus, and
\[ I := \int_A x_3^2 \, dA \]
is the second moment of area of the beam cross-section \(A\). For a rectangular cross-section of width \(b\) and thickness \(h\), \(I = b h^3 / 12\), again exhibiting the characteristic \(h^3\) scaling of bending stiffness.
Boundary conditions for the beam are again two per end: clamped (\(w = w' = 0\)), simply supported (\(w = w'' = 0\)), free (\(w'' = w''' = 0\)), or sliding (\(w' = w''' = 0\)). The cantilever — clamped at \(x = 0\), free at \(x = L\), under uniform load \(q(x) = q_0\) — admits the closed-form solution
\[ w(x) = \frac{q_0}{24 EI} \, x^2 \, (x^2 - 4 L x + 6 L^2), \]
with maximum deflection \(w_{\max} = w(L) = q_0 L^4 / (8 EI)\) at the free end. This is the simplest worked example that anchors the abstract beam theory; its detailed derivation is the subject of the in-class exercise (Exercise 4).
6.11.6 Edge of the linear theory
The Kirchhoff–Love plate and Euler–Bernoulli beam models rest on the assumption that the rotation of the mid-surface remains small. When transverse deflections become comparable to the thickness (\(w \sim h\)), the rotation becomes \(\mathcal O(1)\) and the linearisation Equation 6.12 fails — geometrically, even though the in-plane strain remains small (this is precisely the small-strain-vs-small-displacement distinction flagged at the end of §6.6). The next-order theory, retaining quadratic terms in the rotation, is the Föppl–von Kármán plate model — a coupled nonlinear system for \(w\) and an in-plane stress potential. It is mechanically linear (Hooke’s law Equation 6.6 still holds) but geometrically nonlinear. It captures buckling and post-buckling behaviour and is the natural starting point for any analysis of thin-shell instability. We refer to (Howell, Kozyreff, and Ockendon 2009 ch. 6) for derivation and discussion. In the present chapter we remain inside the fully linear regime.
6.12 Linear viscoelasticity and the correspondence principle
So far the constitutive law has been instantaneous: the stress at a material point depends on the current deformation alone. Many real materials — polymers above their glass transition, biological tissue, geological materials such as ice and rock salt, fresh concrete, polymer melts — exhibit time-dependent response. Held under constant load, they continue to deform (creep). Held under constant deformation, the stress required to maintain that deformation decays over time (stress relaxation). They have memory.
Linear viscoelasticity provides the simplest framework that captures these effects while retaining the linearity that makes elasticity tractable. The classical reference is (Lakes 2009).
6.12.1 One-dimensional models: spring and damper
We work in 1D, with scalar stress \(\sigma\) and scalar strain \(\epsilon\). The elementary mechanical analogues are:
- A linear spring with modulus \(E\) — the Hooke element — obeying \(\sigma = E \epsilon\).
- A linear damper with relaxation time \(\tau\) — the viscous element — obeying \(\sigma = E \tau \, \dot \epsilon\), where the dot denotes \(d/dt\).
The damper is parameterised here through \(E\) and a time scale \(\tau\) (rather than through a separate viscosity coefficient), so that all parameters carry the same units as in elasticity and the limits \(\tau \to 0\) (purely elastic) and \(\tau \to \infty\) (rigid) have a clean interpretation.
Two arrangements give the canonical linear viscoelastic models:
Maxwell model — spring and damper in series. The strain decomposes as \(\epsilon = \epsilon_s + \epsilon_d\), with \(\epsilon_s = \sigma / E\) and \(\dot \epsilon_d = \sigma / (E \tau_M)\). Differentiating and adding:
\[ \dot \epsilon = \frac{\dot \sigma}{E} + \frac{\sigma}{E \tau_M}. \tag{6.18}\]
Kelvin–Voigt model — spring and damper in parallel. The stress is shared, \(\sigma = E \epsilon + E \tau_{KV} \dot \epsilon\), equivalently
\[ \sigma = E \, ( \epsilon + \tau_{KV} \dot \epsilon ). \tag{6.19}\]
The time constants \(\tau_M\) (Maxwell relaxation time) and \(\tau_{KV}\) (Kelvin–Voigt retardation time) characterise the rate at which the material “forgets” its history.
6.12.2 Creep and stress relaxation
The two canonical experiments are:
Creep — apply a sudden constant stress \(\sigma(t) = \sigma_0 H(t)\) (\(H\) the Heaviside function) and observe the strain \(\epsilon(t)\).
For Maxwell, integrating Equation 6.18 with \(\dot \sigma = 0\) for \(t > 0\) and the initial elastic jump \(\epsilon(0^+) = \sigma_0 / E\): \[ \epsilon_M(t) = \frac{\sigma_0}{E} \left( 1 + \frac{t}{\tau_M} \right) \qquad (t > 0). \] Strain grows linearly without bound — the Maxwell material is a fluid on long time scales.
For Kelvin–Voigt, solving Equation 6.19 as a first-order ODE for \(\epsilon\) with \(\sigma = \sigma_0\) and \(\epsilon(0) = 0\): \[ \epsilon_{KV}(t) = \frac{\sigma_0}{E} \left( 1 - e^{-t/\tau_{KV}} \right). \] Strain saturates at the elastic value \(\sigma_0 / E\) — the Kelvin–Voigt material is a delayed solid.
Stress relaxation — apply a sudden constant strain \(\epsilon(t) = \epsilon_0 H(t)\) and observe the stress.
For Maxwell, with \(\dot \epsilon = 0\) for \(t > 0\): \[ \sigma_M(t) = E \epsilon_0 \, e^{-t/\tau_M}. \] Stress decays exponentially to zero — relaxation is the diagnostic of Maxwell behaviour.
For Kelvin–Voigt, the constitutive law gives \(\sigma_{KV}(t) = E \epsilon_0\) for \(t > 0\) (the damper contribution is impulsive at the strain jump and can be handled via distributions). No relaxation.
These four two-line calculations summarise the qualitative phenomenology of linear viscoelasticity. Real materials are typically modelled as combinations of multiple Maxwell and Kelvin–Voigt elements (the generalised Maxwell or Prony series representations); the behaviours interpolate between the bounded-creep / no-relaxation pole of Kelvin–Voigt and the unbounded-creep / full-relaxation pole of Maxwell.
6.12.3 The correspondence principle
Both Maxwell and Kelvin–Voigt constitutive laws are linear ODEs in \(t\). Taking the Laplace transform with respect to time, with \(\hat \sigma(s) := \int_0^\infty \sigma(t) e^{-st} dt\) and similarly for \(\hat \epsilon(s)\), and assuming homogeneous initial conditions:
\[ \begin{aligned} \text{Maxwell:} \qquad &\hat \sigma(s) = \hat E_M(s) \, \hat \epsilon(s), &\qquad \hat E_M(s) &= \frac{E s \tau_M}{1 + s \tau_M}, \\ \text{Kelvin–Voigt:} \qquad &\hat \sigma(s) = \hat E_{KV}(s) \, \hat \epsilon(s), &\qquad \hat E_{KV}(s) &= E (1 + s \tau_{KV}). \end{aligned} \tag{6.20}\]
In Laplace space, the viscoelastic constitutive law has exactly the form of a linear elastic constitutive law, with the constant Young’s modulus \(E\) replaced by an \(s\)-dependent transformed modulus \(\hat E(s)\). This is the correspondence principle:
Any boundary value problem of linear isotropic viscoelasticity, with time-independent boundaries and homogeneous initial conditions, can be solved by:
- Taking the Laplace transform in time. The constitutive law becomes elastic with modulus \(\hat E(s)\).
- Solving the resulting elastic problem in \(s\)-space. All the machinery of Chapter 6 — Navier equation, biharmonic statics, Kirchhoff–Love plate, Euler–Bernoulli beam — applies term-for-term.
- Inverting the Laplace transform to recover the time-dependent solution.
The implication is that the entire chapter — every equation, every boundary value problem — has an immediate viscoelastic counterpart obtained by replacing \(E \mapsto \hat E(s)\) and Laplace-inverting the result. No additional theory is needed; only an appropriate modulus law and a Laplace-transform table.
A worked instance: applying the correspondence principle to the cantilever beam \(EI w'''' = q\) with a Maxwell material gives the creep response of the cantilever as a single inverse Laplace transform. This is the subject of the optional Part 3 of Exercise 4.
For the systematic treatment — including non-isotropic moduli, the Boltzmann hereditary integral form, and the connection between \(\hat E(s)\) and the time-domain relaxation modulus \(E(t)\) — we refer to (Lakes 2009) and to the perturbation-methods background in (Holmes 2013).
6.13 List of Symbols
| Symbol | Type | Description | SI Unit |
|---|---|---|---|
| \(l\), \(\delta l\) | scalar | length of rod / change in length | \([\mathrm{m}]\) |
| \(a\) | scalar | diameter of rod | \([\mathrm{m}]\) |
| \(A\), \(\delta A\) | scalar | cross-sectional area / change in area | \([\mathrm{m}^2]\) |
| \(P\) | vector | axial load/force | \([\mathrm{kg}\ \mathrm{m}\ \mathrm{s}^{-2}]\) |
| \(\epsilon\) | scalar | strain | \([-]\) |
| \(\sigma\) | scalar/tensor | stress | \([\mathrm{m}^{-1}\ \mathrm{kg}\ \mathrm{s}^{-2}]\) |
| \(E\) | scalar | Young’s modulus | \([\mathrm{m}^{-1}\ \mathrm{kg}\ \mathrm{s}^{-2}]\) |
| \(\nu\) | scalar | Poisson’s ratio | \([-]\) |
| \(\rho\) | scalar | density | \([\mathrm{kg}\ \mathrm{m}^{-3}]\) |
| \(\mathbf{v}\) | vector | velocity | \([\mathrm{m}\ \mathrm{s}^{-1}]\) |
| \(p\) | scalar | pressure | \([\mathrm{m}^{-1}\ \mathrm{kg}\ \mathrm{s}^{-2}]\) |
| \(\mathbf{\zeta}\) | vector | material (reference) coordinates | \([\mathrm{m}]\) |
| \(\mathbf{x}\) | vector | spatial (deformed) coordinates | \([\mathrm{m}]\) |
| \(F\) | tensor | deformation gradient | \([-]\) |
| \(Q\) | tensor | rotation tensor | \([-]\) |
| \(U\) | tensor | right stretch tensor | \([-]\) |
| \(V\) | tensor | left stretch tensor | \([-]\) |
| \(B\) | tensor | left Cauchy-Green tensor | \([-]\) |
| \(C\) | tensor | right Cauchy-Green tensor | \([-]\) |
| \(\mathbf{w}\) | vector | displacement field | \([\mathrm{m}]\) |
| \(H\) | tensor | displacement gradient | \([-]\) |
| \(D\) | tensor | small strain tensor | \([-]\) |
| \(D^{(d)}\) | tensor | deviatoric strain tensor | \([-]\) |
| \(\lambda\) | scalar | Lamé’s first parameter | \([\mathrm{m}^{-1}\ \mathrm{kg}\ \mathrm{s}^{-2}]\) |
| \(\mu\) | scalar | shear modulus | \([\mathrm{m}^{-1}\ \mathrm{kg}\ \mathrm{s}^{-2}]\) |
| \(K\) | scalar | bulk modulus | \([\mathrm{m}^{-1}\ \mathrm{kg}\ \mathrm{s}^{-2}]\) |
| \(\delta_{ij}\) | tensor | Kronecker delta | \([-]\) |
| \(\mathbf{n}\) | vector | outward normal vector | \([-]\) |
| \(\mathbf{w}_1\) | vector | solenoidal displacement | \([\mathrm{m}]\) |
| \(\mathbf{w}_2\) | vector | irrotational displacement | \([\mathrm{m}]\) |
| \(a_1\) | scalar | S-wave speed squared | \([\mathrm{m}^2\ \mathrm{s}^{-2}]\) |
| \(a_2\) | scalar | P-wave speed squared | \([\mathrm{m}^2\ \mathrm{s}^{-2}]\) |
| \(\mathbf{I}\) | tensor | identity tensor | \([-]\) |
| \(\nabla\) | operator | nabla operator | \([\mathrm{m}^{-1}]\) |
| \(\Delta\) | operator | Laplacian operator | \([\mathrm{m}^{-2}]\) |
| \(tr(\cdot)\) | operator | trace of a tensor | \([-]\) |
| \(\epsilon\) | scalar | kinematic small parameter (§6.6) | \([-]\) |
| \(\eta\) | scalar | geometric small parameter \(h/L\) (§6.11) | \([-]\) |
| \(h\) | scalar | plate thickness / beam depth | \([\mathrm{m}]\) |
| \(L\) | scalar | in-plane characteristic length | \([\mathrm{m}]\) |
| \(K_b\) | scalar | plate bending stiffness | \([\mathrm{kg}\ \mathrm{m}\ \mathrm{s}^{-2}]\) |
| \(M_{\alpha\beta}\) | tensor | plate moment resultants | \([\mathrm{kg}\ \mathrm{s}^{-2}]\) |
| \(I\) | scalar | second moment of area of beam cross-section | \([\mathrm{m}^4]\) |
| \(EI\) | scalar | beam bending stiffness | \([\mathrm{kg}\ \mathrm{m}^3\ \mathrm{s}^{-2}]\) |
| \(\Delta_2\) | operator | 2D Laplacian on plate mid-surface | \([\mathrm{m}^{-2}]\) |
| \(q\) | scalar | distributed transverse load (per area / per length) | varies |
| \(\tau_M\) | scalar | Maxwell relaxation time | \([\mathrm{s}]\) |
| \(\tau_{KV}\) | scalar | Kelvin–Voigt retardation time | \([\mathrm{s}]\) |
| \(s\) | scalar | Laplace transform variable | \([\mathrm{s}^{-1}]\) |
| \(\hat E(s)\) | scalar | Laplace-transformed elastic modulus | \([\mathrm{m}^{-1}\ \mathrm{kg}\ \mathrm{s}^{-2}]\) |
{tbl-colwidths=“[15,20,45,20]”}
6.14 Further reading
The chapter is supported by four primary references. We list them here with short pointers to where each is most useful.
Gonzalez & Stuart (2008), A First Course in Continuum Mechanics (Gonzalez and Stuart 2001). Primary reference for §§6.1–6.10. Detailed treatment of deformation kinematics, polar decomposition, the spd-square-root construction, material symmetry and objectivity, and the small-displacement reduction.
Howell, Kozyreff & Ockendon (2009), Applied Solid Mechanics (Howell, Kozyreff, and Ockendon 2009). Chapter 4 covers an alternative derivation of the Navier equation and standard analytical boundary-value problems. Chapter 6 is the primary reference for §6.11: full asymptotic derivation of the Kirchhoff–Love plate, Euler–Bernoulli beam, and the Föppl–von Kármán extension.
Holmes (2013), Introduction to Perturbation Methods (Holmes 2013). Background on the asymptotic methodology used in §§6.6 and 6.11. Useful for students who want a systematic treatment of dominant-balance arguments and matched asymptotic expansions.
Lakes (2009), Viscoelastic Materials (Lakes 2009). Primary reference for §6.12. Treats spring–damper models, the Boltzmann hereditary integral, the correspondence principle, and the connection between time-domain and Laplace-domain descriptions.