2.3 Mathematical description of internal forces in solids

 

Our next objective is to outline the mathematical formulas that describe internal and external forces acting on a solid.  Just as there are many different strain measures, there are several different definitions of internal force.  We shall see that internal forces can be described as a second order tensor, which must be symmetric.  Thus, internal forces can always be quantified by a set of six numbers, and the various different definitions are all equivalent.

 

 

 

2.3.1 Surface traction and internal body force

 

Forces can be applied to a solid body in two ways, as illustrated in the figure 

 

1. A force can be applied to its boundary: examples include fluid pressure, wind loading, or forces arising from contact with another solid. 

 

2. The solid can be subjected to body forces, which act on the interior of the solid.  Examples include gravitational loading, or electromagnetic forces.

 

These forces are quantified using the surface traction vector, and the body force vector, respectively.   These are defined as follows:

The surface traction vector t at a point on the surface represents the force acting on the surface per unit area of the deformed solid. Formally, let dA be an element of area on a surface.  Suppose that dA is subjected to a force $dP$, as shown in the figure.  Then

$t=\underset{dA\to 0}{\lim }{\displaystyle \frac{dP}{dA}}$

 

The resultant force acting on any portion S of the surface of the deformed solid is

                                                                 $P={\displaystyle \underset{S}{\int }tdA}$

Surface traction, like `true stress,’ should be thought of as acting on the deformed solid.

 

The traction vector is often resolved into  which are components acting normal and tangential to a surface, as shown in the figure. The normal component is referred to as the normal traction, and the tangential component is known as the shear traction.

 

Formally, let n denote a unit vector normal to the surface.  Then

$t_n=\left(t\cdot n\right)n{t}_t=t-t_n$

 

The body force vector denotes the external force acting on the interior of a solid, per unit mass. Formally, let dV denote an infinitesimal volume element within the deformed solid, and let $\rho$ denote the mass density (mass per unit deformed volume).  Suppose that the element is subjected to a force $dP$, as shown in the figure.  Then

$b={\displaystyle \frac{1}{\rho }\underset{dV\to 0}{\lim }{\displaystyle \frac{dP}{dV}}}$

The resultant body force acting on any volume V  within the deformed solid is

                                                               $P={\displaystyle \underset{V}{\int }\rho bdV}$

 

 

 

2.3.2 Traction acting on planes within a solid

 

Every plane in the interior of a solid is subjected to a distribution of traction.  To see this, consider a loaded, solid, body in static equilibrium.  Imagine cutting the solid in two, as illustrated in the figure. The two parts of the solid must each be in static equilibrium.  This is possible only if forces act on the planes that were created by the cut.  We quantify these forces by means of the internal traction vector T(n), which represents the force per unit area acting on an internal plane of a solid. The notation T(n) shows that the internal traction depends on the normal to the internal plane, denoted by n.

 

Formally, let dA be an element of area in the interior of the solid, with normal n.  Suppose that the material on the underside of dA is subjected to a force $d{P}^{(n)}$ across the plane dA, as shown in the figure. Then

$T(n)=\underset{dA\to 0}{\lim }{\displaystyle \frac{d{P}^{(n)}}{dA}}$

Note that internal traction is the force per unit area of the deformed solid, like `true stress.’  The traction vector has the following properties 

 

· The resultant force acting on any internal volume V with boundary surface A  within a deformed solid is

                                                       $P={\displaystyle \underset{A}{\int }T(n)dA+}{\displaystyle \underset{V}{\int }\rho bdV}$

The first term is the resultant force acting on the internal surface A, the second term is the resultant body force acting on the interior V.

 

 

· Newton’s third law (every action has an equal and opposite reaction) requires that

$T(-n)=-T(n)$

To see this, note that the forces acting on planes separating two adjacent volume elements in a solid must be equal and opposite, as shown in the figure.

 

 

· Traction acting on different planes passing through the same point are related, in order to satisfy Newton’s second law (F=ma).  Specifically, let $\left\{e_1,e_2,e_3\right\}$ be a Cartesian basis.  Let $T_i(e_1)$, $T_i(e_2)$, $T_i(e_3)$ denote the components of traction acting on planes with normal vectors in the $e_1$, $e_2$, and $e_3$ directions, respectively.  Then, the traction components $T_i(n)$ acting on a surface with normal n are given by

$T_i(n)=T_i(e_1)n_1+T_i(e_2)n_2+T_i(e_3)n_3$

where $n_i$ are the components of n.

 

To see this, consider the forces acting on the infinitessimal tetrahedron shown in the figure.  The base and sides of the tetrahedron have normals in the $-e_2$, $-e_1$ and $-e_3$ directions.  The fourth face has normal n.  Suppose the volume of the tetrahedron is dV, and let $d{A}_1$,  $d{A}_2$,  $d{A}_3$,  $d{A}^{(n)}$ denote the areas of the faces.  Assume that the material within the tetrahedron has mass density $\rho$ and is subjected to a body force b. Let a denote the acceleration of the center of mass of the tetrahedron. Then, F=ma for the tetrahedron requires that

                        $T(n)d{A}^{(n)}+T(-e_1)d{A}_1+T(-e_2)d{A}_2+T(-e_3)d{A}_3+\rho bdV=\rho dVa$

Recall that $T(-e_i)=-T(e_i)$ and divide through by $d{A}^{(n)}$:

                     $T(n)-T(e_1){\displaystyle \frac{d{A}_1}{d{A}^{(n)}}-T(e_2){\displaystyle \frac{d{A}_2}{d{A}^{(n)}}-T(e_3){\displaystyle \frac{d{A}_3}{d{A}^{(n)}}+\rho b{\displaystyle \frac{dV}{d{A}^{(n)}}=\rho {\displaystyle \frac{dV}{d{A}^{(n)}}a}}}}}$

Finally, let $d{A}^{(n)}\to 0$.  We can show (see Appendix E) that

                            ${\displaystyle \frac{d{A}_1}{d{A}^{(n)}}=n_1{\displaystyle \frac{d{A}_2}{d{A}^{(n)}}=n_2{\displaystyle \frac{d{A}_3}{d{A}^{(n)}}=n_3\underset{d{A}^{(n)}\to 0}{\lim }{\displaystyle \frac{dV}{d{A}^{(n)}}=0}}}}$

So

                                              $T(n)=T(e_1)n_1+T(e_2)n_2+T(e_3)n_3$

or, using index notation

                                              $T_i(n)=T_i(e_1)n_1+T_i(e_2)n_2+T_i(e_3)n_3$

The significance of this result is that the tractions acting on planes with normals in the $e_1$, $e_2$, and $e_3$ directions completely characterize the internal forces that act at a point.  Given these tractions, we can deduce the tractions acting on any other plane.  This leads directly to the definition of the Cauchy stress tensor in the next section.

 

 

 

2.3.3 The Cauchy (true) stress tensor

 

Consider a solid which deforms under external loading. Let $\left\{e_1,e_2,e_3\right\}$ be a Cartesian basis.  Let $T_i(e_1)$, $T_i(e_2)$, $T_i(e_3)$ denote the components of traction acting on planes with normals in the $e_1$, $e_2$, and $e_3$ directions, respectively, as outlined in the preceding section. 

 

Define the components of the Cauchy stress tensor ${\sigma }_{ij}$ by

                                    $\begin{array}{c}{\sigma }_{ij}=T_j(e_i)\\ \equiv \left\{\begin{array}{l}{\sigma }_{11}=T_1(e_1){\sigma }_{12}=T_2(e_1){\sigma }_{13}=T_3(e_1)\\ {\sigma }_{21}=T_1(e_2){\sigma }_{22}=T_2(e_2){\sigma }_{23}=T_3(e_2)\\ {\sigma }_{31}=T_1(e_3){\sigma }_{32}=T_2(e_3){\sigma }_{33}=T_3(e_3)\end{array}\right.\end{array}$

Then, the traction  $T_i(n)$ acting on any plane with normal n follows as

                                                $T(n)=n\sigma \text{or }{T}_i(n)=n_j{\sigma }_{ji}$

 

To see this, recall the last result from the preceding section

                                              $T_i(n)=T_i(e_1)n_1+T_i(e_2)n_2+T_i(e_3)n_3$

and substitute for $T_i(e_j)$ in terms of the components of the Cauchy stress tensor

                                              $T_i(n)={\sigma }_{1i}{n}_1+{\sigma }_{2i}{n}_2+{\sigma }_{3i}{n}_3=n_j{\sigma }_{ji}$

 

 

The Cauchy stress tensor completely characterizes the internal forces acting in a deformed solid.  The physical significance of the components of the stress tensor is illustrated in the figure. ${\sigma }_{ji}$ represents the ith component of traction acting on a plane with normal in the $e_j$ direction.

 

 

Note that Cauchy stress represents force per unit area of the deformed solid.  In elementary strength of materials courses it is called `true stress,’ for this reason.

 

 

HEALTH WARNING: Some texts define stress as the transpose of the definition used here, so that $T(n)=\sigma n\text{or }{T}_i(n)={\sigma }_{ij}{n}_j$.  In this case the first index for each stress component denotes the direction of traction, while the second denotes the normal to the plane.  We will see later that Cauchy stress is always symmetric, so there is no confusion if you use the wrong definition.  But some stress measures are not symmetric (see below) and in this case you need to be careful to check which convention the author has chosen.

 

 

 

2.3.4 Other stress measures $–$ Kirchhoff, Nominal and Material stress tensors

 

Cauchy stress ${\sigma }_{ij}$ (the actual force per unit area acting on an actual, deformed solid) is the most physical measure of internal force.  Other definitions of stress often appear in constitutive equations, however. 

 

The other stress measures regard forces as acting on the undeformed solid.  Consequently, to define them we must know not only what the deformed solid looks like, but also what it looked like before deformation.  The deformation is described by a displacement vector $u(x)$ and the associated deformation gradient

                                                  $F=I+u\nabla F_{ij}={\delta }_{ij}+{\displaystyle \frac{\partial u_i}{\partial x_j}}$

as outlined in Section 2.2. In addition, let $J=\det (F)$.

 

We then define the following stress measures

 

· Kirchhoff stress  $\tau =J\sigma {\tau }_{ij}=J{\sigma }_{ij}$

 

· Nominal (First Piola-Kirchhoff) stress  $S=J{F}^{-1}\sigma S_{ij}=J{F}_{ik}^{-1}{\sigma }_{kj}$

 

· Material (Second Piola-Kirchhoff) stress  $\Sigma =J{F}^{-1}\sigma F^{-T}{\Sigma }_{ij}=J{F}_{ik}^{-1}{\sigma }_{kl}{F}_{jl}^{-1}$

 

The inverse relations are also useful $–$ the one for Kirchhoff stress is obvious $–$ the others are

                     $\sigma ={\displaystyle \frac{1}{J}FS{\sigma }_{ij}={\displaystyle \frac{1}{J}{F}_{ik}{S}_{kj}\sigma ={\displaystyle \frac{1}{J}F\Sigma F^T{\sigma }_{ij}={\displaystyle \frac{1}{J}{F}_{ik}^{}{\Sigma }_{kl}{F}_{jl}^{}}}}}$

 

The Kirchhoff stress has no obvious physical significance, but the quantity ${\tau }_{ij}{D}_{ij}$ (where D is the stretch rate tensor) represents the rate of work done by stress per unit undeformed volume, which is why the Kirchhoff stress is useful. 

The nominal stress tensor can be regarded as the internal force per unit undeformed area acting within a solid, as follows

 

1. Visualize an element of area dA in the deformed solid, with normal n, which is subjected to a force $d{P}^{(n)}$ by the internal traction in the solid;

 

2. Suppose that the element of area dA has started out as an element of area $d{A}_0$ with normal $n_0$ in the undeformed solid, as shown in the figure.

 

3. Then, the force $d{P}^{(n)}$ is related to the nominal stress by $d{P}_j^{(n)}=d{A}_0{n}_i^0{S}_{ij}$

 

To see this, note that one can show (see Appendix E) that

                                         $dAn=J{F}^{-T}d{A}_0{n}_{0}dA{n}_i^{}=J{F}_{ki}^{-1}{n}_k^{0}d{A}_0$

Recall that the Cauchy stress is defined so that

                                                            $d{P}_i^{(n)}=dA{n}_j{\sigma }_{ji}$

Substituting for $dA{n}_j$ and rearranging shows that

                                              $d{P}_i^{(n)}=Jd{A}_0{n}_k^0\left(F_{kj}^{-1}{\sigma }_{ji}\right)=d{A}_0{n}_k^0{S}_{ki}$

 

 

The material stress tensor can also be visualized as force per unit undeformed area, except that the forces are regarded as acting within the undeformed solid, rather than on the deformed solid.  Specifically

 

1. The infinitesimal force $d{P}^{(n)}$ is assumed to behave like an infinitesimal material fiber in the solid, in the sense that it is stretched and rotated just like an small vector dx in the solid;

 

2. This means that we can define a (fictitious) force in the reference configuration $d{P}^{(n0)}$ that is related to $d{P}^{(n)}$ by $Fd{P}^{(n0)}=d{P}^{(n)}$ or $F_{ij}d{P}_j{}^{(n0)}=d{P}_i{}^{(n)}$;

 

3. This fictitious force is related to material stress by  $d{P}_i^{(n0)}=d{A}_0{n}_j^0{\Sigma }_{ji}$.

 

To see this, substitute into the expression relating $d{P}^{(n)}$ to nominal stress to see that

                                                         $F_{ik}d{P}_k^{(n0)}=d{A}_0{n}_j^0{S}_{ji}$

Finally multiply through by $F_{li}^{-1}$, note $F_{li}^{-1}{F}_{ik}={\delta }_{lk}$, and rearrange to see that

                                                $d{P}_l^{(n0)}=d{A}_0{n}_j^0{S}_{ji}{F}_{li}^{-1}=d{A}_0{n}_j^0{\Sigma }_{jl}$

where we have noted that ${\Sigma }_{jl}=S_{ji}{F}_{li}^{-1}$

 

In practice, it is best not to try to attach too much physical significance to these stress measures.  Cauchy stress is the best physical measure of internal force $–$ it is the force per unit area acting inside the deformed solid.  The other stress measures are best regarded as generalized forces (in the sense of Lagrangian mechanics), which are work-conjugate to particular strain measures.  This means that the stress measure multiplied by the time derivative of the strain measure tells you the rate of work done by the forces.  When setting up any mechanics problem, we always work with conjugate measures of motion and forces.

 

Specifically, we shall show later that the rate of work $\dot{W}$ done by stresses acting on a small material element with volume $d{V}_0$ in the undeformed solid (and volume $dV$ in the deformed solid) can be computed as

                                    $\dot{W}=D_{ij}{\sigma }_{ij}dV=D_{ij}{\tau }_{ji}d{V}_0={\dot{F}}_{ij}{S}_{ji}d{V}_0={\dot{E}}_{ij}{\Sigma }_{ji}d{V}_0$

where $D_{ij}$ is the stretch rate tensor, ${\dot{F}}_{ij}$ is the rate of change of deformation gradient, and ${\dot{E}}_{ij}$ is the rate of change of Lagrange strain tensor.  Note that Cauchy stress (and also Kirchhoff stress) is not conjugate to any convenient strain measure $–$ this is the main reason that nominal and material stresses need to be defined.  The nominal stress is conjugate to the deformation gradient, while the material stress is conjugate to the Lagrange strain tensor. 

 

 

 

2.3.5 Stress measures for infinitesimal deformations

 

For a problem involving infinitesimal deformation (where shape changes are characterized by the infinitesimal strain tensor and rotation tensor) all the stress measures defined in the preceding section are approximately equal

                                                           ${\sigma }_{ij}\approx {\tau }_{ij}\approx S_{ij}\approx {\Sigma }_{ij}$

 

To see this, write the deformation gradient as $F_{ij}={\delta }_{ij}+\partial u_i/\partial x_j$; recall that $J=\det (F)\approx 1+\partial u_k/\partial x_k$, and finally assume that for infinitesimal motions $\partial u_i/\partial x_j<<1$.  Substituting into the formulas relating Cauchy stress, Nominal stress and Material stress, we see that

                            ${\sigma }_{ij}={\displaystyle \frac{1}{J}{F}_{ik}{S}_{kj}\approx {\displaystyle \frac{1}{1+\partial u_p/\partial x_p}\left({\delta }_{ip}+{\displaystyle \frac{\partial u_i}{\partial x_p}}\right)S_{pj}=S_{pj}+\mathrm{...}\approx S_{pj}}}$

The same procedure will show that material stress and Cauchy stress are approximately equal, to within a term of order $\partial u_i/\partial x_j<<1$

 

 

 

 

2.3.6 Principal Stresses and directions

 

For any stress measure, the principal stresses ${\sigma }_i$ and their directions $n^{(i)}$, with i=1..3 are defined such that

                                  $n^{(i)}\sigma ={\sigma }_i{n}^{(i)}\text{or }{n}_j^{(i)}{\sigma }_{jk}={\sigma }_i{n}_k^{(i)}(\text{no sum on }i)$

 

Clearly,

1. The principal stresses are the left eigenvalues of the stress tensor
2. The principal stress directions are the left eigenvectors of the stress tensor

 

The term `left’ eigenvector and eigenvalue indicates that the vector multiplies the tensor on the left. We will see later that Cauchy stress and material stress are both symmetric.  For a symmetric tensor the left and right eigenvalues and vectors are the same.

 

Note that the eigenvectors of a symmetric tensor are orthogonal.  Consequently, the principal Cauchy or material stresses can be visualized as tractions acting normal to the faces of a cube, as show in the figure. The principal directions specify the orientation of this special cube.

 

One can also show that if ${\sigma }_1>{\sigma }_2>{\sigma }_3$, then ${\sigma }_1$ is the largest normal traction acting on any plane passing through the point of interest, while ${\sigma }_3$ is the lowest.  This is helpful in defining damage criteria for brittle materials, which fail when the stress acting normal to a material plane reaches a critical magnitude.

 

In the same vein, it can be shown that the largest shear stress acts on the plane with unit normal vector $m_{\text{shear}}=-(m_1+m_3)/\sqrt{2}$ (at 45^o to the $m_1$ and $m_3$ axes as shown in the figure, and has magnitude ${\tau }_{\text{max}}=({\sigma }_1-{\sigma }_3)/2$.  This observation is useful for defining yield criteria for metal polycrystals, which begin to deform plastically when the shear stress acting on a material plane reaches a critical value.

 

 

 

2.3.7 Hydrostatic and Deviatoric Stress; von Mises effective stress

 

Given the Cauchy stress tensor $\text{σ}$, the following may be defined:

 

· The Hydrostatic stress is defined as ${\sigma }_h=\text{trace(}\sigma )/3\equiv {\sigma }_{kk}/3$

 

· The Deviatoric stress tensor is defined as ${{\sigma }^{\prime }}_{ij}={\sigma }_{ij}-{\sigma }_h{\delta }_{ij}$

 

· The Von-Mises effective stress is defined as ${\sigma }_e=\sqrt{{\displaystyle \frac{3}{2}{{\sigma }^{\prime }}_{ij}{{\sigma }^{\prime }}_{ij}}}$

 

The hydrostatic stress is a measure of the pressure exerted by a state of stress.  Pressure acts so as to change the volume of a material element.

 

The deviatoric stress is a measure of the shearing exerted by a state of stress. Shear stress tends to distort a solid, without changing its volume.

 

The Von-Mises effective stress can be regarded as a uniaxial equivalent of a multi-axial stress state.  It is used in many failure or yield criteria.  Thus, if a material is known to fail in a uniaxial tensile test (with ${\sigma }_{11}$ the only nonzero stress component) when ${\sigma }_{11}={\sigma }_{crit}$, it will fail when ${\sigma }_e={\sigma }_{crit}$ under multi-axial loading (with several ${\sigma }_{ij}\ne 0$ )

 

The hydrostatic stress and von Mises stress can also be expressed in terms of principal stresses as

                                     $\begin{array}{l}{\sigma }_h=\left({\sigma }_1+{\sigma }_2+{\sigma }_3\right)/3\\ {\sigma }_e=\sqrt{{\displaystyle \frac{1}{2}\left\{{\left({\sigma }_1-{\sigma }_2\right)}^2+{\left({\sigma }_1-{\sigma }_3\right)}^2+{\left({\sigma }_2-{\sigma }_3\right)}^2\right\}}}\end{array}$

 

The hydrostatic and von Mises stresses are invariants of the stress tensor $–$ they have the same value regardless of the basis chosen to define the stress components.

 

 

 

2.3.8 Stresses near an external surface or edge $–$ boundary conditions on stresses

 

Note that at an external surface at which tractions are prescribed, some components of stress are known.  Specifically, let n denote a unit vector normal to the surface, and let t denote the traction (force per unit area) acting on the surface.  Then the Cauchy stress at the surface must satisfy

                                                                  $n_i{\sigma }_{ij}=t_j$

 

For example, suppose that a surface with normal in the $e_2$ direction is subjected to no loading, as shown in the figure.  Then (noting that $n_i={\delta }_{i2}$ ) it follows that ${\sigma }_{2i}=0$. In addition, two of the principal stress directions must be parallel to the surface; the third (with zero stress) must be perpendicular to the surface.

 

The stress state at an edge is even simpler.  Suppose that surfaces with normals in the $e_2$ and $e_1$ are traction free.  Then ${\sigma }_{1i}={\sigma }_{2i}=0$, so that 6 stress components are known to be zero.