.. _linear_elastic_ivp_bvp:

#############################################
Linear Elastic Initial/Boundary Value Problem
#############################################

************************************
Basic Equations of Linear Elasticity
************************************

Having reviewed some relevant nonlinearities in the context of a simple structural element in Chapter :numref:`nonlinear_behavior`, let us begin to generalize our problem description to encompass a larger group of continuous bodies. We begin this development by first reviewing the basic equations of *linear elasticity*, where we assume small motions and linear material behavior. This discussion will provide the basis for a more general notational framework in the next section, where we will remove the kinematic restriction to small motions and also allow the material to behave in an inelastic manner.

The notation we will use in this section is summarized in :numref:`linear_elastic_ibvp-fig-notation`, where we have depicted a solid body positioned in the three dimensional Euclidean space, or :math:`\mathbb{R}^3`. The set of spatial points :math:`\mathbf{x}` defining the body is denoted by :math:`\Omega`, and we consider the boundary :math:`\partial \Omega` to be subdivided into two regions :math:`\Gamma _u` and :math:`\Gamma _{\sigma}`, where Dirichlet and Neumann boundary conditions will be specified as discussed below. We assume that these regions obey the following:

.. math::
   :label: linear_elastic_ivbp:eq:01

   \begin{gathered}
    \Gamma _{u} \cup \Gamma _{\sigma} = \partial \Omega \\ 
    \Gamma _{u} \cap \Gamma _{\sigma} = \emptyset.
   \end{gathered}

The unknown, or independent, variable in this problem is :math:`\mathbf{u}`, the vector-valued displacement which in general depends upon :math:`\mathbf{x} \in \Omega` and time :math:`t`.

.. _linear_elastic_ibvp-fig-notation:

.. figure:: ../_static/figures/linear_elastic_ivp_bvp-fig01.png
   :align: center
   :scale: 75 %

   Notation for the linear elastic initial/boundary value problem.

*******************
Equations of Motion
*******************

At any point :math:`\Omega` the following statement of local linear momentum balance must hold:

.. math::
   :label: linear_elastic_ivbp:eq:02

   \nabla \cdot \mathbf{T} + \mathbf{f} = \rho \frac{\partial ^2 \mathbf{u}}{\partial t^2}  .

Note that :math:`\nabla \cdot \mathbf{T}` denotes the divergence operator applied to :math:`\mathbf{T}`, the Cauchy stress tensor. The vector :math:`\mathbf{f}` denotes the distributed body force in :math:`\Omega`, with units of force per volume, and :math:`\rho` denotes the mass density, which need not be uniform. :eq:`linear_elastic_ivbp:eq:02` represents the balance of linear momentum in direct notation.  Balance of angular momentum is enforced within the domain by requiring that the Cauchy stress tensor is symmetric. We will frequently employ **index notation** in the work that follows. Toward that end, :eq:`linear_elastic_ivbp:eq:02` can be expressed as

.. math::
   :label: linear_elastic_ivbp:eq:03

   T_{ij,j} + f_i = \rho \frac{\partial ^2 u_i}{\partial t^2}  ,


where indices :math:`i` and :math:`j` run between 1 and 3 (the spatial directions), and unless otherwise indicated, repeated indices within a term of an expression imply a summation over that index. For example,

.. math::
   :label: linear_elastic_ivbp:eq:04

   T_{ij,j} = \sum_{j=1}^{3} \frac{\partial T_{ij}}{\partial x_j}  .


The notation :math:`\beta _{,j}` indicates partial differentiation with respect to :math:`x_j`.

As indicated above the independent variables are :math:`u_i`, so it is necessary to specify the relation between the displacements and the Cauchy stress. In linear elasticity this is accomplished by two additional equations. The first is the linear strain-displacement relation

.. math::
   :label: linear_elastic_ivbp:eq:05

   \epsilon_{ij} = u_{(i,j)} = \frac{1}{2} (u_{i,j} + u_{j,i}) ,

where :math:`\epsilon_{ij}` is the infinitesimal strain equal to the symmetric part of the **displacement gradient** denoted by :math:`u_{(i,j)}`. The second equation is the linear **constitutive relation** between :math:`T_{ij}` and :math:`\epsilon_{ij}`, which is normally written

.. math::
   :label: linear_elastic_ivbp:eq:06

   T_{ij} = C_{ijkl} \epsilon_{kl}.

Note that :math:`C_{ijkl}` is the fourth-order elasticity tensor, to be discussed further below.

:eq:`linear_elastic_ivbp:eq:05` and :eq:`linear_elastic_ivbp:eq:06` can also be written in direct notation as

.. math::
   :label: linear_elastic_ivbp:eq:07

   \boldsymbol{\epsilon} = \nabla_{s} \mathbf{u} = \frac{1}{2} \left( \nabla \mathbf{u} + \nabla \mathbf{u}^T \right),

where :math:`\nabla_{s}` denotes the symmetric gradient operator defined by :math:`\nabla_{s} \Box = 1/2 \left( \nabla \Box + \nabla \Box^T \right)`, and

.. math::
   :label: linear_elastic_ivbp:eq:08

   \mathbf{T} = \mathbf{C}:\boldsymbol{\epsilon},

where the colon indicates double contraction of the fourth-order tensor :math:`\mathbf{C}` with the second-order tensor :math:`\boldsymbol{\epsilon}`.

The fourth-order elasticity tensor :math:`\mathbf{C}` is ordinarily assumed to possess a number of symmetries, which greatly reduces the number of independent components that describe it. It possesses **major symmetry**, which means :math:`C_{ijkl} = C_{klij}`, and it also possesses **minor symmetries**, meaning for example that :math:`C_{ijkl} = C_{jikl} = C_{jilk} = C_{ijlk}`. Another important property of the elasticity tensor is **positive definiteness**, implying in this context that

.. math::
   :label: linear_elastic_ivbp:eq:09

   A_{ij} C_{ijkl} A_{kl} >0 \quad \text{ for all symmetric tensors } A


.. math::
   :label: linear_elastic_ivbp:eq:10

   \text{and } A_{ij} C_{ijkl} A_{kl} =0 \quad \text{ iff } A=0.

In the most general case, assuming the aforementioned symmetries and no others, the elasticity tensor has 21 independent components. Various material symmetries reduce the number greatly, the most specific case being an **isotropic** material possessing rotational symmetry in all directions. In this case only two independent elastic constants are required to specify :math:`\mathbf{C}`, which under these circumstances can be written as

.. math::
   :label: linear_elastic_ivbp:eq:11

   C_{ijkl} = \lambda \delta_{ij} \delta_{kl} + \mu \left[ \delta_{ik} \delta_{jl} + \delta_{il} \delta_{jk} \right], 

where :math:`\delta_{ij}`, the **Kronecker delta**, satisfies

.. math::
   :label: linear_elastic_ivbp:eq:12

   \delta_{ij} = 
   \begin{cases}
    1 & \quad \text{if } i=j \\
    0 & \quad \text{otherwise},
   \end{cases}

and :math:`\lambda` and :math:`\mu` denote the **Lam\'e parameters** for the material. These can be written in terms of the more familiar **Young's (i.e., elastic) modulus** and **Poisson's ratio** via

.. math::
   :label: linear_elastic_ivbp:eq:13

   \lambda = \frac{E \nu}{(1+\nu)(1-2\nu)}


.. math::
   :label: linear_elastic_ivbp:eq:14

   \mu = \frac{E}{2(1+\nu)}.

The quantity :math:`\mu` is also known as the **shear modulus** for the material.

Substitution of :eq:`linear_elastic_ivbp:eq:07` and :eq:`linear_elastic_ivbp:eq:08` into :eq:`linear_elastic_ivbp:eq:02` gives a partial differential equation for the vector-valued unknown displacement field :math:`\mathbf{u}`. Full specification of the problem with suitable boundary and initial conditions is discussed next.

*******************************
Boundary and Initial Conditions
*******************************

Paralleling earlier discussion of the one-dimensional example, we will consider the possibility of two types of boundary conditions, Dirichlet and Neumann. Dirichlet boundary conditions will be imposed on the region :math:`\Gamma_{u}` in :numref:`linear_elastic_ibvp-fig-notation` as

.. math::
   :label: linear_elastic_ivbp:eq:15

   \mathbf{u} (\mathbf{x},t) = \bar{\mathbf{u}} (\mathbf{x},t) \quad \forall \mathbf{x} \in \Gamma_{u}, \quad t \in (0,T) .

Note that :math:`\bar{\mathbf{u}} (\mathbf{x},t)` denotes a prescribed displacement vector depending on spatial position and time. The simplest and perhaps most common example of such a boundary condition would be a fixed condition, which if imposed throughout the time interval of interest :math:`(0,T)` and for all of :math:`\Gamma_{u}` would imply :math:`\bar{\mathbf{u}} (\mathbf{x},t) = \mathbf{0}`.

The other type of boundary condition is a Neumann, or traction, boundary condition. To write such a condition we must first define the concept of **traction** on a surface. If we use :math:`\mathbf{n}` to denote the outward normal to the surface :math:`\Gamma_{\sigma}` at a point :math:`\mathbf{x} \in \Gamma_{\sigma}`, the traction vector :math:`\mathbf{t}` at :math:`\mathbf{x}` is defined via

.. math::
   :label: linear_elastic_ivbp:eq:16

   \mathbf{t} = \mathbf{T} \cdot \mathbf{n},

or, in index notation,

.. math::
   :label: linear_elastic_ivbp:eq:17

   T_{i} = T_{ij} n_{j}.

Physically this vector represents a force per unit area acting on the external surface at :math:`\mathbf{x}`. A Neumann boundary condition is then written in the current notation as

.. math::
   :label: linear_elastic_ivbp:eq:18

   \mathbf{T} (\mathbf{x},t) \cdot \mathbf{n} (\mathbf{x} )= \bar{\mathbf{t}} (\mathbf{x},t)  \quad \forall \mathbf{x} \in \Gamma_{\sigma}, \quad t \in (0,T) .

Note that :math:`\bar{\mathbf{t}} (\mathbf{x},t)` is the prescribed traction vector field on :math:`\Gamma_{\sigma}` throughout the time interval of interest :math:`(0,T)`.
One could identify several examples of such a boundary condition. An unfixed surface free of any external force would be described by :math:`\bar{\mathbf{t}} = \mathbf{0}`. A surface subject to a uniform pressure loading, :math:`p`, on the other hand, could be described by setting :math:`\bar{\mathbf{t}} (\mathbf{x},t) = -p\mathbf{n}(\mathbf{x})`, where we assume a compressive pressure to be positive.

With these definitions in hand, we recall the restrictions in :eq:`linear_elastic_ivbp:eq:01` on :math:`\Gamma_{u}` and :math:`\Gamma_{\sigma}` and physically interpret them as follows: 1) one must specify either a traction or a displacement boundary condition at every point of :math:`\partial \Omega`; and 2) at each point of :math:`\partial \Omega` one may not specify both the traction and the displacement but must specify one or the other.

In fact these conditions are slightly more stringent than required. The problem remains well-posed if, for each component direction :math:`i`, we specify either the traction component :math:`\bar{t}_i` or the displacement component :math:`\bar{u}_i` at each point :math:`\mathbf{x} \in \partial \Omega`, as long as for a given spatial direction we do not attempt to specify both. In other words, we may specify a displacement boundary condition in one direction at a point while specifying a traction boundary condition in the other. An example of such a case would be the common roller boundary condition, where a point is free to move in a traction-free manner to an interface (i.e., a traction boundary condition) while being constrained from movement in a direction normal to an interface (i.e., a displacement boundary condition). Of course a multitude of other boundary condition permutations could be identified. Thus, while we choose a rather simple boundary condition restriction (summarized by :eq:`linear_elastic_ivbp:eq:01`) for notational simplicity, it is important to realize that many other possibilities exist and require only minor alterations of the methodology we will discuss.

The final important ingredient in our statement of the linear elastic problem is the specification of initial conditions. One may note that our partial differential equation (:eq:`linear_elastic_ivbp:eq:02`) is second order in time; accordingly, two initial conditions are required. In the current context these are the initial conditions on the displacement :math:`\mathbf{u}` and the velocity :math:`\dot{\mathbf{u}}` and can be rather straightforwardly specified as

.. math::
   :label: linear_elastic_ivbp:eq:19

   \mathbf{u} (\mathbf{x},0) =  \mathbf{u}_0 (\mathbf{x}) \quad \text{on } \Omega


.. math::
   :label: linear_elastic_ivbp:eq:20

   \frac{\partial \mathbf{u}}{\partial t} (\mathbf{x},0) =  \mathbf{v}_0 (\mathbf{x}) \quad \text{on } \Omega,

where :math:`\mathbf{u}_0` and :math:`\mathbf{v}_0` are the prescribed initial displacement and velocity fields, respectively.

*********************
Problem Specification
*********************

We now collect the equations and conditions of the past two sections into a single problem statement for the linear elastic system shown in :numref:`linear_elastic_ibvp-fig-notation`. For the elastodynamic case, this problem falls into the category of an **initial/boundary value problem**, since both types of conditions are included in its definition. Our problem is formally stated as follows:

Given the boundary conditions :math:`\bar{**t**}` on :math:`\Gamma_{\sigma} \times (0,T)` and :math:`\bar{**u**}` on :math:`\Gamma_{u} \times (0,T)`, the initial conditions :math:`\mathbf{u} _0` and :math:`\mathbf{v}_0` on :math:`\Omega`, and the distributed body force :math:`\mathbf{f}` on :math:`\Omega \times (0,T)`, find the displacement field :math:`\mathbf{u}` on :math:`\Omega \times (0,T)` such that

.. math::
   :label: linear_elastic_ivbp:eq:21

   \nabla \cdot \mathbf{T} + \mathbf{f} = \rho \frac{\partial ^2 \mathbf{u}}{\partial t^2} \quad \text{on } \Omega \times (0,T),


.. math::
   :label: linear_elastic_ivbp:eq:22

   \mathbf{u}(\mathbf{x},t) = \bar{\mathbf{u}}(\mathbf{x},t) \quad \text{on } \Gamma_{u} \times (0,T),


.. math::
   :label: linear_elastic_ivbp:eq:23

   \mathbf{t}(\mathbf{x},t) = \bar{\mathbf{t}}(\mathbf{x},t) \quad \text{on } \Gamma_{\sigma} \times (0,T),


.. math::
   :label: linear_elastic_ivbp:eq:24

   \mathbf{u}(\mathbf{x},0) = \mathbf{u}_0 (\mathbf{x}) \quad \text{on } \Omega ,


.. math::
   :label: linear_elastic_ivbp:eq:25

   \frac{\partial \mathbf{u}}{\partial t}(\mathbf{x},0) = \mathbf{v}_0 (\mathbf{x}) \quad \text{on } \Omega ,

where the Cauchy stress, :math:`\mathbf{T}`, is given by

.. math::
   :label: linear_elastic_ivbp:eq:26

   \mathbf{T} = \mathbf{C} : ( \nabla _s \mathbf{u} ).

Equations :eq:`linear_elastic_ivbp:eq:21` through :eq:`linear_elastic_ivbp:eq:26` constitute a linear hyperbolic initial/boundary value problem for the independent variable :math:`\mathbf{u}`.

*****************************
The Quasistatic Approximation
*****************************

Before leaving the elastic problem, it is worthwhile to discuss how our problem specification will change if inertial effects are negligible in the equilibrium equations. This special case is often referred to as the **quasistatic assumption** and considerably simplifies specification of the problem.

Simply stated, the quasistatic assumption removes the second temporal derivative of :math:`\mathbf{u}`, i.e., acceleration, from :eq:`linear_elastic_ivbp:eq:21`, thereby also eliminating the need for initial conditions (Equations :eq:`linear_elastic_ivbp:eq:24` and :eq:`linear_elastic_ivbp:eq:25`). Such an approximation is appropriate when the loadings do not vary with time or when they vary over time scales much longer than the periods associated with the fundamental structural modes of :math:`\Omega`.

It is convenient, however, to maintain time in our description of the problem for two reasons: 1) the loadings :math:`\bar{\mathbf{t}}` and :math:`\mathbf{f}` and the displacement condition :math:`\bar{\mathbf{u}}` may still vary with time; and 2) when we consider more general classes of constitutive equations, we may wish to allow time dependence in the stress/strain response, e.g., in creep plasticity. Accordingly, we state below a boundary value problem appropriate for quasistatic response of a linear elastic system.

Given the boundary conditions :math:`\bar{\mathbf{t}}` on :math:`\Gamma_{\sigma} \times (0,T)`, :math:`\bar{\mathbf{u}}` on :math:`\Gamma_{u} \times (0,T)`, and the distributed body force :math:`\mathbf{f}` on :math:`\Omega \times (0,T)`, find the displacement field :math:`\mathbf{u}` on :math:`\Omega \times (0,T)` such that

.. math::
   :label: linear_elastic_ivbp:eq:27

   \nabla \cdot \mathbf{T} + \mathbf{f} = 0 \quad \text{on } \Omega \times (0,T),


.. math::
   :label: linear_elastic_ivbp:eq:28

   \mathbf{u}(\mathbf{x},t) = \bar{\mathbf{u}}(\mathbf{x},t) \quad \text{on } \Gamma_{u} \times (0,T),


.. math::
   :label: linear_elastic_ivbp:eq:29

   \mathbf{t}(\mathbf{x},t) = \bar{\mathbf{t}}(\mathbf{x},t) \quad \text{on } \Gamma_{\sigma} \times (0,T),

where the Cauchy stress, :math:`\mathbf{T}`, is given by

.. math::
   :label: linear_elastic_ivbp:eq:30

   \mathbf{T} = \mathbf{C} : ( \nabla _s \mathbf{u} ).

We note in that given a time :math:`t \in (0,T)`, Equations :eq:`linear_elastic_ivbp:eq:27` through :eq:`linear_elastic_ivbp:eq:30` constitute a linear elliptic boundary value problem governing the independent variable :math:`\mathbf{u}`.
