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
[../]
[]