A mathematical treatment of one-dimensional soil consolidation

Abstract

Terzaghi’s conception of the nature of one-dimensional soil consolidation [1] is shown to lead to a non-linear differential equation. A dimensional analysis of this equation and the boundary conditions of the standard consolidation test [2] gives a more general explanation of a well known linear relationship between the total consolidation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U left-parenthesis t right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>U</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">U\left ( t \right )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> after a time <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t"> <mml:semantics> <mml:mi>t</mml:mi> <mml:annotation encoding="application/x-tex">t</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t Superscript 1 slash 2"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>t</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{t^{1/2}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. By linearizing the equation in a general manner, an expression is obtained for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U left-parenthesis t right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>U</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">U\left ( t \right )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which includes secondary consolidation terms. Two solutions of the linearized equation are obtained; the first for the standard consolidation test and the second for consolidation under a boundary load increasing uniformly with time.