The elastic stored energy of initially strained, or stressed, materials: restrictions and third-order expansions

A large variety of engineering and biological materials have a non-zero internal stress distribution, even in the absence of applied forces. These stresses can arise from thermal expansion or volumetric growth, for example, in the production of the material. There are two approaches to modelling such materials that appear similar but are, in fact, distinct. The first defines a function, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mstyle displaystyle="true" scriptlevel="0"> <mml:mrow> <mml:mrow> <mml:mover> <mml:mi>W</mml:mi> <mml:mo stretchy="false">~</mml:mo> </mml:mover> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi mathvariant="bold">F</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> <mml:mi mathvariant="bold-italic">τ</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:mstyle> </mml:math> , associated with a fixed reference configuration , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:semantics> <mml:mrow> <mml:mrow> <mml:mi class="mathcal" mathvariant="script">B</mml:mi> </mml:mrow> </mml:mrow> </mml:semantics> </mml:math> , say, where each value of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mstyle displaystyle="true" scriptlevel="0"> <mml:mrow> <mml:mi mathvariant="bold-italic">τ</mml:mi> </mml:mrow> </mml:mstyle> </mml:math> corresponds to the initial stress in a different elastic material that occupies <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:semantics> <mml:mrow> <mml:mrow> <mml:mi class="mathcal" mathvariant="script">B</mml:mi> </mml:mrow> </mml:mrow> </mml:semantics> </mml:math> (each with a different elastic constitutive equation, effectively). The second defines a function, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mstyle displaystyle="true" scriptlevel="0"> <mml:mrow> <mml:mrow> <mml:mover> <mml:mi>W</mml:mi> <mml:mo accent="false">¯</mml:mo> </mml:mover> </mml:mrow> <mml:mspace width="thinmathspace"/> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi mathvariant="bold">F</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> <mml:mi mathvariant="bold-italic">τ</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:mstyle> </mml:math> , associated with a single, fixed, initially stressed, elastic material (with a single constitutive equation), where each value of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mstyle displaystyle="true" scriptlevel="0"> <mml:mrow> <mml:mi mathvariant="bold-italic">τ</mml:mi> </mml:mrow> </mml:mstyle> </mml:math> represents the stress in a different configuration of that material. Here, we discuss why stored energy functions of the latter type, and similar functions that are written in terms of an initial strain , need to satisfy some restrictions to avoid unphysical behaviours. To illustrate their need, we perform an asymptotic expansion to prove that these restrictions are required for consistency with strain energy functions of classical third-order weakly nonlinear elasticity.