Applicable for small linearized strains, MOOSE includes an incremental small strain material, [ComputeIncrementalSmallStrain](http://mooseframework.org/docs/doxygen/modules/classComputeIncrementalSmallStrain.html).  This material uses the linearized small strain definition that is appropriate when the gradient of displacement with respect to position is much smaller than unity:

$$ \epsilon = \frac{1}{2} \left( u \nabla + \nabla u \right) \quad when \quad \frac{\partial u}{ \partial x} << 1 $$

For more details on the linearized small strain assumption and derivation, see a Continuum Mechanics text such as Malvern (1969) or [Bower(2012)](http://solidmechanics.org/Text/Chapter2_1/Chapter2_1.php#Sect2_1_7).

As the class name suggests, `ComputeIncrementalSmallStrain` is an incremental formulation.  The stress increment is calculated from the current strain increment at each time step.  In this class, the rotation tensor is defined to be the rank-2 Identity tensor: no rotations are allowed in the model. Stateful properties, including `strain_old` and `stress_old`, are stored. This incremental small strain material is useful as a component of verifying more complex finite incremental strain-stress calculations.

### Input File Use

The incremental small strain material can be used with any incremental elastic stress material, such as [ComputeFiniteStrainElasticStress](http://mooseframework.org/docs/doxygen/modules/classComputeFiniteStrainElasticStress.html). An example portion of an input file using these two materials is shown below:
```yaml
  ## Tensor Mechanics Version
  [Materials]
    [./strain]
      type = ComputeIncrementalSmallStrain
      block = 0
      displacements = 'disp_x disp_y disp_z'
    [../]
    [./stress]
      type = ComputeFiniteElasticStress
      block = 0
    [../]
    [./elasticity_tensor]
      type = ComputeIsotropicElasticityTensor
      block = 0
      poissons_ratio = 0.3
      youngs_modulus = 2.1e9
    [../]
  []
```

Incremental small strain formulations for 1D spherically symmetric and 2D axisymmetric problems are also included in MOOSE; more information on these specialized materials can be found [here](http://mooseframework.com/wiki/PhysicsModules/TensorMechanics/LowerOrderGeometryMaterials/).

#### Input file conversion from Solid Mechanics to Tensor Mechanics
The material combination `ComputeIncrementalSmallStrain`, `ComputeFiniteStrainElasticStress`, and an elasticity tensor material will replace the Solid Mechanics `SolidModel` or `Elastic` option `formulation = linear` when used with the option `large_strain = true`.  An equivalent solid mechanics input file to the Tensor Mechanics input file shown above is:

```yaml
  ## Old Solid Mechanics Version
  [Materials]
    [./linelast]
      type = Elastic
      block = 0
      disp_x = disp_x
      disp_y = disp_y
      disp_z = disp_z
      poissons_ratio = 0.3
      youngs_modulus = 2.1e9
      formulation = Linear
      large_strain = true
    [../]
  []
```